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D

Updated in 14.1

✖
D[f,x]

gives the partial derivative .

✖
D[f,{x,n}]

gives the multiple derivative .

✖
D[f,x,y,…]

gives the partial derivative .

✖
D[f,{x,n},{y,m},…]

gives the multiple partial derivative .

✖
D[f,{{x1,x2,…}}]

for a scalar f gives the vector derivative .

✖
D[f,{array}]

gives an array derivative.

Details and Options

  • D is also known as derivative for univariate functions.
  • By using the character ∂, entered as pd or \[PartialD], with subscripts, derivatives can be entered as follows:
  • D[f,x]∂xf
    D[f,{x,n}]∂{x,n}f
    D[f,x,y]∂x,yf
    D[f,{{x,y}}]∂{{x,y}}f
  • The comma can be made invisible by using the character \[InvisibleComma] or ,.
  • The partial derivative D[f[x],x] is defined as , and higher derivatives D[f[x,y],x,y] are defined recursively as etc.
  • The order of derivatives n and m can be symbolic and they are assumed to be positive integers.
  • The derivative D[f[x],{x,n}] for a symbolic f is represented as Derivative[n][f][x].
  • For some functions f, Derivative[n][f][x] may not be known, but can be approximated by applying N. »
  • New derivative rules can be added by adding values to Derivative[n][f][x]. »
  • For lists, D[{f1,f2,…},x] is equivalent to {D[f1,x],D[f2,x],…} recursively. »
  • D[f,{array}] effectively threads D over each element of array.
  • D[f,{array,n}] is equivalent to D[f,{array},{array},…], where {array} is repeated n times.
  • D[f,{array1},{array2},…] is normally equivalent to First[Outer[D,{f},array1,array2,…]]. »
  • Common array derivatives include:
  • D[f,{{x1,x2,…}}]gradient{D[f,x1],D[f,x2],…}
    D[f,{{x1,x2,…},2}]Hessian{{D[f,x1,x1],D[f,x1,x2],…},{D[f,x2,x1],D[f,x2,x2],…},…}
    D[{f1,f2,…},{{x1,x2,…}}]Jacobian{{D[f1,x1],D[f1,x2],…},
    {D[f2,x1],D[f2,x2],…},…}
  • If f is a scalar and x={x1,…}, then the multivariate Taylor series at x0={x01,…} is given by:
  • ,
  • where fi=D[f,{x,i}]/.{x1x01,…} is an array with tensor rank . »
  • If f and x are both arrays, then D[f,{x}] effectively threads first over each element of f, and then over each element of x. The result is an array with dimensions Join[Dimensions[f],Dimensions[x]]. »
  • VectorSymbol, MatrixSymbol or ArraySymbol can be used to indicate that variables or function values are vectors, matrices or arrays.
  • D can formally differentiate operators such as integrals and sums, taking into account scoped variables as well as the structure of the particular operator.
  • Examples of operator derivatives include:
  • is not scoped by the integral
    is scoped by the integral
    is not scoped by the integral transform
    is scoped by by the integral transform
  • All expressions that do not explicitly depend on the variables given are taken to have zero partial derivative.
  • The setting NonConstants{u1,…} specifies that ui depends on all variables x, y, etc. and does not have zero partial derivative. »

Examples

open allclose all

Basic Examples  (7)Summary of the most common use cases

Derivative with respect to x:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-we3
Out[1]=1

Fourth derivative with respect to x:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-e12
Out[1]=1

Derivative of order n with respect to x:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-py6oh
Out[1]=1

Derivative with respect to x and y:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-r5c
Out[1]=1

Derivative involving a symbolic function f:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-r9e
Out[1]=1

Evaluate derivatives numerically:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-mc4
Out[1]=1

Enter ∂ using pd, and subscripts using :

In[1]:=1
✖
https://wolfram.com/xid/0p2n-u49
Out[1]=1

Scope  (89)Survey of the scope of standard use cases

Basic Uses  (12)

Derivative of an expression with respect to x:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-gvkdq9
Out[1]=1

Second derivative:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-c5s7uv
Out[2]=2

Derivative of an expression at a point:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-dteywp
Out[1]=1

Derivative of a function at a general point x:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-b0dmye
In[2]:=2
✖
https://wolfram.com/xid/0p2n-864zzf
Out[2]=2

This can also be achieved using fluxion notation:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-c9hgli
Out[3]=3

Derivative at the point x==5:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-811p92
Out[4]=4

This can be found more easily using fluxion notation:

In[5]:=5
✖
https://wolfram.com/xid/0p2n-98tmy
Out[5]=5

The third derivative at the point x==-1:

In[6]:=6
✖
https://wolfram.com/xid/0p2n-3j12m2
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0p2n-b70li9
Out[7]=7

Derivative involving symbolic functions:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-ba5n5b
Out[1]=1

Partial derivatives of an expression with respect to x and y:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-mh0wo9
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-jp52py
Out[2]=2

The mixed partial derivative :

In[3]:=3
✖
https://wolfram.com/xid/0p2n-dywtox
Out[3]=3

The mixed partial derivative :

In[4]:=4
✖
https://wolfram.com/xid/0p2n-bj3lw2
Out[4]=4

Differentiate with respect to a compound expression:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-vrev78
Out[1]=1

Differentiate with respect to different compound expressions:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-b22eur
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-eq4hzp
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0p2n-febzdf
Out[3]=3

Derivative of a vector expression:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-q7rhk6
Out[1]=1

Matrix expression:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-dydr0x
Out[2]=2

Derivative of a nested list:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-d45ix8
Out[1]=1

Vector derivative of an expression, also known as the gradient:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-pay3at
Out[1]=1

Second vector derivative, also known as the Hessian:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-5zmt4q
Out[2]=2

Matrix derivative:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-jziy30
Out[1]=1

Create a table of basic derivatives:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-bxb0ht
In[2]:=2
✖
https://wolfram.com/xid/0p2n-r3pr9e

Symbolic Functions  (9)

Derivative of a symbolic function:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-xe3
Out[1]=1

Substitute in a pure function for f to get a particular result:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-w26
Out[2]=2

Derivative of a sum:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-dupsdn
Out[1]=1

Product:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-crhcke
Out[2]=2

Quotient:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-bxzn0g
Out[3]=3

The chain rule for composite functions:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-6xk5v
Out[4]=4

Product rule for three functions:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-emh4gt
Out[1]=1

State the rule using an Inactive derivative:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-jk8hgs
Out[2]=2

Partial derivative of a symbolic function:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-x3f
Out[1]=1

Substitute in for f a pure function in two variables:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-c585uj
Out[2]=2

Derivative of a pure function with respect to non-argument parameters:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-ddu6m1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-bg09k1
Out[2]=2

The result is the function that at point x gives the derivative of with respect to a:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-b3533
Out[3]=3

Local variables are independent from the differentiation variable:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-lynv4y
In[2]:=2
✖
https://wolfram.com/xid/0p2n-dfc8z
Out[2]=2

Derivative of a symbolic function at a point:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-b277u5
In[2]:=2
✖
https://wolfram.com/xid/0p2n-hbhbjw
Out[2]=2

The same, using prime notation:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-gbp6vd
Out[3]=3

Derivative of an inverse function:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-lrv1fn
Out[1]=1

Product rule for derivatives of order n:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-tlatb
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-d4ok86
Out[2]=2

Chain rule:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-d9lxq8
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-c2rv7f
Out[4]=4

Elementary Functions  (6)

Polynomial and rational functions:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-g7ntx
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-doacxq
Out[2]=2

Algebraic functions:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-fi6y01
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-gte3io
Out[2]=2

Trigonometric and inverse trigonometric functions:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-h049nd
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-nf26a3
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0p2n-bry8uk
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-dagw4w
Out[4]=4

Exponential and logarithmic functions:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-g1w71
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-fx6sco
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0p2n-jgo0c0
Out[3]=3

Hyperbolic functions:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-lg34u3
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-8jucb
Out[2]=2

Create a function that turns a list of expressions into a nicely formatted table of derivatives:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-b6bgvo

Create a table of trigonometric derivatives:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-bbul1

Create a table of hyperbolic derivatives:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-nafzby

Special Functions  (8)

The logarithmic derivative of Gamma is the PolyGamma function:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-klncfm
Out[1]=1

Derivatives of Airy functions are given in terms of AiryAiPrime and AiryBiPrime:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-b05ojl
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-bpa8cf
Out[2]=2

The derivative of Zeta has a closed-form expression at the origin:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-d7cdh7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-esz8mo
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0p2n-ikq777
Out[3]=3

Special functions with elementary derivatives:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-bg6g61
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-cq7dx2
Out[2]=2

Special functions with derivatives expressed in terms of the same functions:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-evp3uh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-0qxy3
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0p2n-ccefoa
Out[3]=3

Derivative of JacobiSN:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-nfcqwl
Out[1]=1

Derivative of JacobiCD:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-rfww7
Out[2]=2

Derivative of LogIntegral:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-j4tpku
Out[1]=1

Derivative of ExpIntegralEi:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-8m3cx
Out[2]=2

Derivative of order n for SinIntegral:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-sckxk
Out[3]=3

Create a table of special function derivatives:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-9nq1kg
In[2]:=2
✖
https://wolfram.com/xid/0p2n-i698j1

Piecewise and Generalized Functions  (8)

Derivative of a piecewise function:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-exx50
Out[1]=1

Derivative of a ConditionalExpression:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-qo8y5t
Out[1]=1

Convert a symbolic function into a piecewise function over the reals to differentiate it:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-3vinq6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-kt7mm
Out[2]=2

Compute the piecewise derivative over a finite range:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-bxzoxm
Out[3]=3

Classical derivatives of pointwise-defined engineering functions:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-11mwjc
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-mxfyv4
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0p2n-yr0309
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-d3e67v
Out[4]=4

Distributional derivatives of generalized functions:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-ebc850
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-fcc0df
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0p2n-uvadeu
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-njwaym
Out[4]=4

Derivative of RealAbs:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-d6gjix
Out[1]=1

RealSign:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-bcs26l
Out[2]=2

Their counterparts on the complex plane are nowhere differentiable:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-liy7kl
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-kq5o71
Out[4]=4

Derivative of Floor:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-drrycs
Out[1]=1

Ceiling:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-oqzy8l
Out[2]=2

Derivatives of functions defined procedurally:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-65aury
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-mpaj9d
Out[2]=2

Implicitly Defined Functions  (3)

D threads over Equal to compute the derivatives of implicit functions:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-cvysog
In[2]:=2
✖
https://wolfram.com/xid/0p2n-hqswci
Out[2]=2

Compute a partial derivative for an implicit function of two variables:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-cx1jfi
In[2]:=2
✖
https://wolfram.com/xid/0p2n-df5u2p
Out[2]=2

Find partials for implicit functions defined by a system of equations:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-7se2c4
In[2]:=2
✖
https://wolfram.com/xid/0p2n-rr7ngf
Out[2]=2

Vector-Valued Functions  (5)

Derivative of a list:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-dh5pon
Out[1]=1

Second derivative:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-pg8dgj
Out[2]=2

The first derivative to a vector-valued function at a general value t:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-w1j0us
In[2]:=2
✖
https://wolfram.com/xid/0p2n-3mlslp
Out[2]=2

Computing the same using prime notation:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-rrsom4
Out[3]=3

The third derivative at t==0:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-ew9tqp
Out[4]=4

Computing the same using prime notation:

In[5]:=5
✖
https://wolfram.com/xid/0p2n-qavwvt
Out[5]=5

The derivative of a matrix:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-rhh9eh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-nmly9o
Out[2]=2

The fourth derivative:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-8i4mw
Out[3]=3

The derivative of a vector-valued function stored as a SparseArray:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-27s6yd
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-0gocl1
Out[2]=2

Convert the result to a normal array:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-eqabwh
Out[3]=3

The derivative of matrix represented as a SymmetrizedArray object:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-jmtqzg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-roj9r6
Out[2]=2

Convert the result to a normal matrix:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-0hkini
Out[3]=3

Vector Argument Functions  (6)

Gradient of a scalar function:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-la66uh
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0p2n-l5rhlu
Out[3]=3

Hessian matrix:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-ez9yej

Jacobian of a vector-valued function:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-hcq2gz
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-dwsq3i

Second-order derivative tensor:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-bbtpdn
Out[3]=3

Compute the derivative of the determinant with respect to the original matrix:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-lds3jw
Out[1]=1

The gradient of a vector-valued function stored as a SparseArray:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-31h1ah
Out[1]=1

The result is another SparseArray, containing only the nonzero entries:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-swghwe
Out[2]=2

Convert the result to a normal matrix:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-xzt9bu
Out[3]=3

Hessian computed as a SparseArray:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-tpvir
In[2]:=2
✖
https://wolfram.com/xid/0p2n-r7t3r
Out[2]=2

The gradient can also be computed as a SparseArray, but in this case it is effectively dense:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-colzae
Out[3]=3

Jacobian computed as a SparseArray:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-b615oi
In[2]:=2
✖
https://wolfram.com/xid/0p2n-buds2k
Out[2]=2

Symbolic Array Arguments and Functions  (8)

Derivative of a symbolic vector-valued function with respect to a scalar argument:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-mms52b
Out[1]=1

Derivative of a symbolic matrix–valued function with respect to a scalar argument:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-ea3pen
Out[1]=1

Derivative of a symbolic array–valued function with respect to a scalar argument:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-e25xw5
Out[1]=1

Derivatives of scalar-valued functions with respect to symbolic vector arguments:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-ce1jav
Out[1]=1

Real symbolic vector argument:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-p75na
Out[2]=2

Derivatives of scalar-valued functions with respect to symbolic matrix arguments:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-buc0gf
Out[1]=1

Real symbolic matrix argument:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-2jxp8
Out[2]=2

Derivative of a scalar-valued function with respect to a symbolic array argument:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-b25t7
Out[1]=1

Derivatives of symbolic array–valued functions with respect to symbolic array arguments:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-czr5fi
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-rjbusx
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0p2n-bb20q0
Out[3]=3

Derivative of a composition of symbolic array functions:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-bnruex
Out[1]=1

Integrals and Integral Transforms  (6)

Differentiate unevaluated integrals:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-bzxbg2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-khrzxz
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0p2n-eq5hsd
Out[3]=3

Fourier transforms:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-lxa92u
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-wktkup
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0p2n-c006c3
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-jste3o
Out[4]=4

Laplace transforms:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-4fxxb7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-nv13ku
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0p2n-4ibksr
Out[3]=3

Convolutions:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-v7xkfw
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-z72cll
Out[2]=2

Differentiate the Inactive form of an integral to get the fundamental theorem of calculus:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-wyrtkz
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-fnog6a
Out[2]=2

A more general form of the fundamental theorem:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-b8hlds
Out[3]=3

Differentiate an inactive FourierTransform:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-cy1bv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-2l8e5f
Out[2]=2

Verify the formal result:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-fmzmvn
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-tlned4
Out[4]=4

Sums and Summation Transforms  (4)

Differentiate an unevaluated sum:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-dh6goc
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-m4czst
Out[2]=2

Differentiation with respect to the dummy variable gives zero:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-u6pt6k
Out[3]=3

Discrete convolution:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-m0yfd2
Out[1]=1

Differentiate the Inactive form of a sum:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-dnnsf0
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-e7x3la
Out[2]=2

ZTransform:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-bf87j5
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-bpokz9
Out[4]=4

Differentiate an inactive GeneratingFunction:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-b6al9m
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-800tz
Out[2]=2

Verify the formal result:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-hef2p2
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-t11q0
Out[4]=4

Indexed Differentiation  (9)

Differentiate with respect to an indexed variable, introducing KroneckerDelta factors:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-vcaibd
Out[1]=1

Use Inactive to prevent expansion of the sum:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-zl8sv4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-l5nig9
Out[2]=2

Summation indices will be renamed if needed, to avoid name ambiguities:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-3gzpv9
Out[1]=1

Differentiate an inactive table with respect to an indexed variable:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-51j9xp
Out[1]=1

Activate the result to get the explicit vector result:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-ob4mnk
Out[2]=2

Differentiate an inactive table twice with respect to an indexed variable:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-jdjast
Out[1]=1

In this case only the j^(th) entry is nonzero:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-9pokk8
Out[2]=2

Use any notation for indexed variables in sums and tables:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-1b4mtn
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-5u7px9
Out[2]=2

Differentiate with respect to a symbolic table of indexed variables:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-jmv09i
Out[1]=1

Activating the result gives the explicit gradient:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-x9ysge
Out[2]=2

Differentiate twice with respect to a symbolic table of indexed variables, introducing a dummy index:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-kpdw2z
Out[1]=1

Replace symbolic variables with explicit values:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-xzrfl8
Out[2]=2

Use symbolic vector differentiation of another symbolic vector:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-0fu71v
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-8ns8a3
Out[2]=2

Vector differentiation of a vector with respect to itself gives the identity matrix:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-ynrs00
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-x5o31a
Out[4]=4

Functions Defined by Derivatives  (5)

Define the derivative with prime notation:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-lob5h

This rule is used to evaluate the derivative:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-vqxpl
Out[2]=2

Define the derivative at a point:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-z7fz9
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-hia7bd
Out[2]=2

Define the second derivative:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-h6epw8
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-jky7ea
Out[2]=2

Prescribe values and derivatives of f and g:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-ilz1om
In[2]:=2
✖
https://wolfram.com/xid/0p2n-by783c
In[3]:=3
✖
https://wolfram.com/xid/0p2n-b8mwki

Find the derivative of the composition at x=3:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-iy09qv
Out[4]=4

Define a partial derivative with Derivative:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-jn0
In[2]:=2
✖
https://wolfram.com/xid/0p2n-54
Out[2]=2

Options  (1)Common values & functionality for each option

NonConstants  (1)

Differentiate with y considered as depending on x:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-bomqy0
Out[1]=1

Solve for the derivative of y to effect implicit differentiation:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-i3emvf
Out[2]=2

Applications  (47)Sample problems that can be solved with this function

Geometry of the Derivative  (5)

The derivative gives the slope of the tangent line at a point:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-g7c3vp
In[2]:=2
✖
https://wolfram.com/xid/0p2n-cvgy5g

For small displacements h from the base point π, the tangent line gives an excellent approximate of f:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-g6tw97

The tangent and f are visually indistinguishable from each other over a small, and only a small, plot range:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-mx8g8r
Out[4]=4

The derivative gives the limit of the slope of the secant line connecting {x,f[x]} to {x+h,f[x+h]}:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-l8b02u
In[2]:=2
✖
https://wolfram.com/xid/0p2n-ebq0wz
Out[2]=2

Visualize the process for the point {1,f[1]}:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-bn2czj
In[4]:=4
✖
https://wolfram.com/xid/0p2n-l5oobd
Out[4]=4

Find an equation for the tangent line to a function:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-c9jf2u

General equation for the tangent line at x=a:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-jei7f3

Tangent line at x=4:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-eoxxel
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-frl9iu
Out[4]=4

Find an equation for the normal line to a function:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-jaf7o3

General equation for the normal line at x=a:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-opbvu7

Normal line at x=1:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-go0sdv
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-jswwqf
Out[4]=4

Find equations for the tangent lines to a function that pass through a point:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-b5rlg9

General equation for the tangent line at x=a:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-e0ixay

Find the tangents to f[x] that pass through (0,-4):

In[3]:=3
✖
https://wolfram.com/xid/0p2n-c44e99
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-dzmw3
Out[4]=4

Characterization of Functions  (5)

Find the turning points on a plane curve:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-cb5nx1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-bn0sqd
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0p2n-dcruky
Out[3]=3

Find the critical points of a function:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-n21l8l
In[2]:=2
✖
https://wolfram.com/xid/0p2n-gk8ily
Out[2]=2

By the second derivative test, these are all maxima or minima:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-v2ng1d
Out[3]=3

Visualize the critical points:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-k22ye9
Out[4]=4

Find all values of c that satisfy the Mean Value theorem on an interval:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-ejatfo
In[2]:=2
✖
https://wolfram.com/xid/0p2n-cm0mmx
In[3]:=3
✖
https://wolfram.com/xid/0p2n-ioxew9
Out[3]=3

Define the secant line from a to b:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-5lw12s

Define the tangent lines associated with the two values of c:

In[5]:=5
✖
https://wolfram.com/xid/0p2n-rzg937

Visualize the two tangent lines parallel to the secant line along with the original function:

In[6]:=6
✖
https://wolfram.com/xid/0p2n-cqamri
Out[6]=6

Use the first derivative to characterize a function:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-9v9f4

Find the critical points of a function:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-efyrik
Out[2]=2

Find where the function is increasing:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-h6vew
Out[3]=3

Find where the function is decreasing:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-hih8pr
Out[4]=4

Visualize the results:

In[5]:=5
✖
https://wolfram.com/xid/0p2n-elkhk1
Out[5]=5

Use the second derivative to characterize a function:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-bl4ozl

Find the inflection points of a function:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-frt4pc
Out[2]=2

Find where the function has positive concavity:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-94bat
Out[3]=3

Find where the function has negative concavity:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-b6m8oj
Out[4]=4

Visualize the results:

In[5]:=5
✖
https://wolfram.com/xid/0p2n-cpvnlt
Out[5]=5

Relation to Integration  (2)

Perform the change of variable t=x^2 in an integral:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-erey
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-b4fkw
Out[2]=2

Verify the results of symbolic integration:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-u3dsl
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-0etun
Out[2]=2

Multivariate and Vector Calculus  (6)

Find the critical points of a function of two variables:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-wle784
In[2]:=2
✖
https://wolfram.com/xid/0p2n-6rwfnq
Out[2]=2

Compute the signs of and the determinant of the second partial derivatives:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-eqxtcj
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-lr5do9
Out[4]=4

By the second derivative test, the first two points—red and blue in the plot—are minima and the third—green in the plot—is a saddle point:

In[5]:=5
✖
https://wolfram.com/xid/0p2n-k30p10
Out[5]=5

Find the curvature of a circular helix with radius r and pitch c:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-gi5dzw
In[2]:=2
✖
https://wolfram.com/xid/0p2n-egbe2w
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0p2n-22xwb
In[4]:=4
✖
https://wolfram.com/xid/0p2n-bpbn1p
Out[4]=4

Obtain the same result using ArcCurvature:

In[5]:=5
✖
https://wolfram.com/xid/0p2n-cbxp3d
Out[5]=5

Compute a univariate Taylor series by hand:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-f05swh
Out[1]=1

Compute a multivariate Taylor series by hand:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-7ia3tp
Out[2]=2

Write a function to automate the process:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-znwo10

Recompute the above using the new function:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-nlkltz
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0p2n-dxpsay
Out[5]=5

The gradient vector can be computed by finding the partial derivatives of a function:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-p3id6r
Out[1]=1

Find the gradient vector of the function :

In[2]:=2
✖
https://wolfram.com/xid/0p2n-cgbo1n
In[3]:=3
✖
https://wolfram.com/xid/0p2n-irx8mg
Out[3]=3

Visualize the direction of the gradient vector using a unit vector representation:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-d2n2ra
In[5]:=5
✖
https://wolfram.com/xid/0p2n-8ej5td
Out[5]=5

The curl of a vector field on the plane can be computed by subtracting the derivatives of its components:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-cfoy1y
Out[1]=1

Find the curl of the vector field :

In[2]:=2
✖
https://wolfram.com/xid/0p2n-donovb
In[3]:=3
✖
https://wolfram.com/xid/0p2n-dd0uhu
Out[3]=3

Visualize the 2D curl as the net "rotation" of the vector field at a point, with red and green representing clockwise and counterclockwise curl, respectively, and radius proportional to the magnitude of rotation:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-c0iubz
Out[4]=4

The divergence of a vector field can be computed by summing the derivatives of its components:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-bk5ztx
Out[1]=1

Find the divergence of a 2D vector field:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-leh081
In[3]:=3
✖
https://wolfram.com/xid/0p2n-bhwvd
Out[3]=3

Visualize 2D divergence as the net "flow" of the vector field at a point, with red and green representing outflow and inflow, respectively, and radius proportional to the magnitude of the flow:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-d7xvf0
Out[4]=4

Differential Equations  (6)

Construct the differential equation satisfied by an implicit function y[x]:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-f06lzw
In[2]:=2
✖
https://wolfram.com/xid/0p2n-bog1s0
Out[2]=2

Use D to specify ordinary and partial differential equations:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-b6i7dp
In[2]:=2
✖
https://wolfram.com/xid/0p2n-cj66l
Out[2]=2

These can be solved using DSolve:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-dpca0x
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-egq6bm
Out[4]=4

Define a wave equation in two spatial variables:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-bpci1l

Define initial values for the function and its first time derivative:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-baqlqv

Specify boundary conditions:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-jexkca

Solve the system using DSolve:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-k7gljh

Extract a few terms from the Inactive sum:

In[5]:=5
✖
https://wolfram.com/xid/0p2n-byvlbi
Out[5]=5

The two-dimensional wave executes periodic motion in the vertical direction:

In[6]:=6
✖
https://wolfram.com/xid/0p2n-dzejet
Out[6]=6

Specify a Laplacian operator using D:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-sq6djy

Specify homogeneous Dirichlet boundary conditions:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-tzp4z

Find the 4 smallest eigenvalues and eigenfunctions of the operator in a unit disk:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-646c3s
In[4]:=4
✖
https://wolfram.com/xid/0p2n-4ir027

Visualize the eigenfunctions:

In[5]:=5
✖
https://wolfram.com/xid/0p2n-dvcswe
Out[5]=5

Specify an integro-differential equation using D:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-la45bu

Obtain the general solution:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-bpg8tp
Out[2]=2

Specify an initial condition to obtain a particular solution:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-mqxej5
Out[3]=3

Plot the solutions for different values of a:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-k9lub5
Out[4]=4

Find a second-degree polynomial solution to the differential equation:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-bhp5yc
In[2]:=2
✖
https://wolfram.com/xid/0p2n-fx0p2d
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0p2n-cnvd51
Out[3]=3

Rates of Change  (5)

The height of a projectile at time t is given by:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-e2nv2

Compute the velocity at t:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-rxn19
Out[2]=2

Compute the acceleration at t:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-e4owv9
Out[3]=3

Find when the projectile reaches its maximum height:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-bccblv
Out[4]=4

Find the maximum height of the projectile:

In[5]:=5
✖
https://wolfram.com/xid/0p2n-e6zg6n
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0p2n-fv5a0
Out[6]=6

The area of a circle as a function of time is given by:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-bprd52

Compute the rate of change of area:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-gbdpjf
Out[2]=2

Find the rate of change of area at a radius of 10 m if the radius increases at a rate of 5m/s:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-232ym5
Out[3]=3

The position of a particle is given by:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-4sq5h

Compute the velocity, acceleration, jerk, snap (jounce), crackle and pop of the particle:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-bbu3pj
In[3]:=3
✖
https://wolfram.com/xid/0p2n-gg8hjx
Out[3]=3

The total resistance in a circuit of two resistors connected in parallel is given by:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-ef24xb
Out[1]=1

Calculate RT for the given values of R1 and R2:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-gbinls
Out[2]=2

Find the rate of change of the total resistance:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-dlk6uo
Out[3]=3

Calculate the rate of change of the total resistance with the given values:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-cv6hva
Out[4]=4

Volume of a cube in terms of side length l is given by:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-qd02zy

Surface area of a cube is given by:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-bbfkvf

Compute the rate of change of the volume of a cube with respect to surface area using the chain rule:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-sqcdis
Out[3]=3

Solve for l in terms of surface area and substitute that into the result:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-c9o10f
Out[4]=4

Implicit Functions  (3)

Find an equation for the tangent line to an implicit function at (1,):

In[6]:=6
✖
https://wolfram.com/xid/0p2n-c2c977
In[7]:=7
✖
https://wolfram.com/xid/0p2n-w13me
Out[7]=7

Compute the slope at x=1:

In[8]:=8
✖
https://wolfram.com/xid/0p2n-dz6ah7
Out[8]=8

Tangent line at x=1:

In[9]:=9
✖
https://wolfram.com/xid/0p2n-kmposq
Out[9]=9

Visualize the tangent:

In[10]:=10
✖
https://wolfram.com/xid/0p2n-fq9qo7
Out[10]=10

Find the points where the slope of an implicit function equals :

In[1]:=1
✖
https://wolfram.com/xid/0p2n-q2olt1
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-fpcma
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0p2n-bgkhdl
Out[3]=3

The relationship between functions is given by:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-bdw1dl
Out[1]=1

Find the derivative of z[t]:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-ifbmak
Out[2]=2

Calculate z'[t] with the given values:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-c0ggwf
Out[3]=3

Optimization  (3)

Find the maximum area of a rectangular fence of 2000 ft., bordered on one side by a barn:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-i4ud41
In[2]:=2
✖
https://wolfram.com/xid/0p2n-botnsc

Compute the area in terms of width:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-34geg
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-mbr2bb
Out[4]=4

Find the maximum:

In[5]:=5
✖
https://wolfram.com/xid/0p2n-b8hicp
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0p2n-b2tqxy
Out[6]=6

By the second derivative test, this value is a true maximum:

In[7]:=7
✖
https://wolfram.com/xid/0p2n-85khle
Out[7]=7

Alternately, compute the area in terms of length:

In[8]:=8
✖
https://wolfram.com/xid/0p2n-m8qgc9
Out[8]=8
In[9]:=9
✖
https://wolfram.com/xid/0p2n-ee18xj
Out[9]=9
In[10]:=10
✖
https://wolfram.com/xid/0p2n-bpuz16
Out[10]=10

Visualize how the area changes as the length changes:

In[11]:=11
✖
https://wolfram.com/xid/0p2n-ccuf53
Out[11]=11

Find the shortest distance from a curve to the point (1,5):

In[1]:=1
✖
https://wolfram.com/xid/0p2n-6n3gy
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-f1oft5

Compute the distance in terms of y:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-fhl9a9
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-cai6bf
Out[4]=4

Find the minimum point:

In[5]:=5
✖
https://wolfram.com/xid/0p2n-vok2q
Out[5]=5

By the second derivative test, this is a minimum:

In[6]:=6
✖
https://wolfram.com/xid/0p2n-y8eeu5
Out[6]=6

Visualize how the distance changes with position:

In[7]:=7
✖
https://wolfram.com/xid/0p2n-dghb8w
Out[11]=11

Find the dimensions of a lidless, cylindrical can with the least material that can hold up to 2 L of water:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-cgsm3z
In[2]:=2
✖
https://wolfram.com/xid/0p2n-hr93h8

Compute the height in terms of the radius, using the volume constraint:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-khu5o4
Out[3]=3

Compute the surface area in terms of the radius:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-im30sj
Out[4]=4

The radius corresponding to the minimum surface area:

In[5]:=5
✖
https://wolfram.com/xid/0p2n-cch57q
Out[5]=5

By the second derivative test, this is a minimum:

In[6]:=6
✖
https://wolfram.com/xid/0p2n-2mvyyw
Out[6]=6

Compute the radius, height and surface area of the minimum configuration:

In[7]:=7
✖
https://wolfram.com/xid/0p2n-tioc3
Out[7]=7

Visualize how the dimensions vary with radius:

In[8]:=8
✖
https://wolfram.com/xid/0p2n-fvyn7c
Out[11]=11

L'Hôpital's Rule  (3)

Find the limit of the ratio of two functions as x0:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-osc9v
In[2]:=2
✖
https://wolfram.com/xid/0p2n-4bnpl

Directly solving the limit leads to an indeterminate form of type :

In[3]:=3
✖
https://wolfram.com/xid/0p2n-jlkc08
Out[3]=3

L'Hôpital's rule can be used because in an interval around , both and are defined, and :

In[4]:=4
✖
https://wolfram.com/xid/0p2n-oegoa2
Out[4]=4

Indeed, and are continuous, and , so can be computed trivially:

In[5]:=5
✖
https://wolfram.com/xid/0p2n-5kda2r
Out[5]=5

Verify the result using Limit:

In[6]:=6
✖
https://wolfram.com/xid/0p2n-v0lbro
Out[6]=6

Visualize the two functions and their ratio:

In[7]:=7
✖
https://wolfram.com/xid/0p2n-czcmwx
Out[7]=7

Find the limit of the ratio of two functions as x∞:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-c44wwk
In[2]:=2
✖
https://wolfram.com/xid/0p2n-dmngfi

Directly solving the limit leads to an indeterminate form of type :

In[3]:=3
✖
https://wolfram.com/xid/0p2n-ixyl29
Out[3]=3

L'Hôpital's rule can be used because for all , both and are defined, and :

In[4]:=4
✖
https://wolfram.com/xid/0p2n-nqti82
Out[4]=4

However, using the first derivatives also leads to an indeterminate form:

In[5]:=5
✖
https://wolfram.com/xid/0p2n-8895o3
Out[5]=5

The second derivatives are constant and obviously satisfy the conditions of L'Hôpital's rule:

In[6]:=6
✖
https://wolfram.com/xid/0p2n-oogzz9
Out[6]=6

Hence can be computed trivially:

In[7]:=7
✖
https://wolfram.com/xid/0p2n-2zoas3
Out[7]=7

Verify the result using Limit:

In[8]:=8
✖
https://wolfram.com/xid/0p2n-646yr
Out[8]=8

Find the limit of the product of two functions as x0:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-fs7e86
In[2]:=2
✖
https://wolfram.com/xid/0p2n-f6mrtp

Directly solving the limit leads to an indeterminate form of type 0×∞:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-cczhud
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-dzryw5
Out[4]=4

Note that is not defined:

In[5]:=5
✖
https://wolfram.com/xid/0p2n-4xtsar
Out[5]=5

However, exists and is positive for all , and it also exists and is negative for all :

In[6]:=6
✖
https://wolfram.com/xid/0p2n-6gpsfw
Out[6]=6

As is clearly defined for all real , L'Hôpital's rule can be applied in the form:

In[7]:=7
✖
https://wolfram.com/xid/0p2n-mzipow
Out[7]=7

The quotient in the right-hand limit gives a continuous expression whose limit is simple to compute:

In[8]:=8
✖
https://wolfram.com/xid/0p2n-qk5zu3
Out[8]=8
In[9]:=9
✖
https://wolfram.com/xid/0p2n-r6figz
Out[9]=9
In[10]:=10
✖
https://wolfram.com/xid/0p2n-okyysa
Out[10]=10

Symbolic Array Calculus  (6)

Approximate the variance for a perturbed vector:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-hj0dom

Order zero approximation:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-bl41ry

Compare with the exact value:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-ilqgfx
Out[4]=4

Order one approximation:

In[5]:=5
✖
https://wolfram.com/xid/0p2n-bbq5cd
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0p2n-il1bi9

Compare with the exact value:

In[8]:=8
✖
https://wolfram.com/xid/0p2n-exd2ap
Out[8]=8

Order two approximation:

In[9]:=9
✖
https://wolfram.com/xid/0p2n-fv4g02
Out[9]=9
In[10]:=10
✖
https://wolfram.com/xid/0p2n-iefg8z

Since the second derivative does not depend on , the order two approximation equals the exact value:

In[11]:=11
✖
https://wolfram.com/xid/0p2n-b1mu81
Out[11]=11

Approximate the determinant of a perturbed matrix:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-e1g4ef

Order zero approximation:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-f53ye9

Compare with the exact value:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-bd1th
Out[3]=3

Order one approximation:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-dd2mk1
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0p2n-e06g12

Compare with the exact value:

In[6]:=6
✖
https://wolfram.com/xid/0p2n-j8wd1j
Out[6]=6

Order two approximation:

In[7]:=7
✖
https://wolfram.com/xid/0p2n-3ygib
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0p2n-b6yuq7

Compare with the exact value:

In[9]:=9
✖
https://wolfram.com/xid/0p2n-euf9ux
Out[9]=9

Derive a least-squares solution for data given as a list of pairs :

In[1]:=1
✖
https://wolfram.com/xid/0p2n-zbx89
Out[1]=1

Find the vector of vertical deviations for the data:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-hfi5q
Out[2]=2

Define the sum of squares of the vertical deviations for the data:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-f4zwc4
Out[3]=3

Set up the least-squares equations:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-feg7h8
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0p2n-jfkuoy
Out[5]=5

Generate some data:

In[6]:=6
✖
https://wolfram.com/xid/0p2n-krd8p6

Solve the least-squares problem for this data:

In[7]:=7
✖
https://wolfram.com/xid/0p2n-td08t
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0p2n-bknhxs
Out[8]=8

Find GammaDistribution parameters that best fit the given data using the maximum likelihood method:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-r959rr
In[2]:=2
✖
https://wolfram.com/xid/0p2n-d8vicu
Out[2]=2

Maximize the log-likelihood function :

In[3]:=3
✖
https://wolfram.com/xid/0p2n-pds2w7
In[4]:=4
✖
https://wolfram.com/xid/0p2n-1mvhdk
Out[4]=4

Compute the gradient of :

In[5]:=5
✖
https://wolfram.com/xid/0p2n-dntaii
Out[5]=5

Find a zero of the gradient, with replaced by :

In[6]:=6
✖
https://wolfram.com/xid/0p2n-zc222s
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0p2n-407hj1
Out[7]=7

Visualize the result:

In[8]:=8
✖
https://wolfram.com/xid/0p2n-trdxv2
Out[8]=8

Compare with the result computed using EstimatedDistribution:

In[9]:=9
✖
https://wolfram.com/xid/0p2n-0lvc75
Out[9]=9

Find an optimality condition for a portfolio optimization problem with the expected return and standard deviation :

In[1]:=1
✖
https://wolfram.com/xid/0p2n-gl52to

The goal is to maximize when the vector of asset weights satisfies Total[x]=1. The constraint can be used to represent where the unconstrained vector variable consists of the first coordinates of :

In[2]:=2
✖
https://wolfram.com/xid/0p2n-en1jbs

The maximum occurs at a critical point of :

In[3]:=3
✖
https://wolfram.com/xid/0p2n-erc7jq
Out[3]=3

Express the condition in terms of :

In[4]:=4
✖
https://wolfram.com/xid/0p2n-bf6j70
Out[4]=4

Compute the gradient of the log-likelihood function of the linear regression model represented by the equation , where are normally distributed random variables with mean zero and variance :

In[1]:=1
✖
https://wolfram.com/xid/0p2n-j42hww

The log-likelihood function is given by:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-zxu0l
Out[2]=2

Compute :

In[3]:=3
✖
https://wolfram.com/xid/0p2n-ezzjdm
Out[3]=3

Express the result in terms of :

In[4]:=4
✖
https://wolfram.com/xid/0p2n-hg3hjc
Out[4]=4

Compute :

In[5]:=5
✖
https://wolfram.com/xid/0p2n-0yff
Out[5]=5

Other Applications  (3)

Compute the coefficients of a power series:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-ltk277
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-zky2k
Out[2]=2

Derive a closed form for by differentiating with respect to at :

In[1]:=1
✖
https://wolfram.com/xid/0p2n-b4doap
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-f9q4hy
Out[2]=2

Now integrate first and then differentiate with respect to at :

In[3]:=3
✖
https://wolfram.com/xid/0p2n-d1ual
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-jxjfln
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0p2n-ju1am7
Out[5]=5

The final result:

In[6]:=6
✖
https://wolfram.com/xid/0p2n-hr52n4
Out[6]=6

Verify the result:

In[7]:=7
✖
https://wolfram.com/xid/0p2n-xy5lme
Out[7]=7

Derive a closed form for by differentiating w.r.t. at :

In[1]:=1
✖
https://wolfram.com/xid/0p2n-f5ni86
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-dbfe6o
Out[2]=2

Compute and then differentiate:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-cuztj8
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-dibvt3
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0p2n-fdfpdr
Out[5]=5

The result:

In[6]:=6
✖
https://wolfram.com/xid/0p2n-frj8ta
Out[6]=6

Verify the result:

In[7]:=7
✖
https://wolfram.com/xid/0p2n-8rub4r
Out[7]=7

Properties & Relations  (23)Properties of the function, and connections to other functions

The derivative of a function is defined as a limit:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-w6b8h
In[2]:=2
✖
https://wolfram.com/xid/0p2n-ev44gs
Out[2]=2

The Limit of DifferenceQuotient is the derivative D:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-1bqnh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-jdw7j8
Out[2]=2

D is the inverse of Integrate:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-dh6trp
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-oethy
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0p2n-zbmmp5
Out[3]=3

The fundamental theorem of calculus:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-nd2pg
Out[1]=1

Differentiation inside of Integrate:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-ngleep
Out[1]=1

D returns formal results in terms of Derivative:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-bmbyk9
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-bth8bn
In[3]:=3
✖
https://wolfram.com/xid/0p2n-j9kd36
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-clhb8g

D differentiates expressions with respect to a given variable:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-qqe0jo
Out[1]=1

Derivative is an operator and returns pure-function results:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-sfmp1l
Out[2]=2

The derivative of a function at a point may not be available in closed form:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-dfw9qq
Out[1]=1

An approximation to the derivative can be obtained using N:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-htm85
Out[2]=2

D[f,{array1},…] is essentially equivalent to First[Outer[D,{f},array1,…]]:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-ydxxu0
Out[1]=1

If f and a are arrays, Dimensions[D[f,{a}]==Join[Dimensions[f],Dimensions[a]]:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-w194bn
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-ynwo2m
Out[2]=2

D[f,{{x1,x2,…,xn}}] is effectively equivalent to Grad[f,{x1,x2,…,xn}]:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-om34tq
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-g7vm64
Out[2]=2

Div[{f1,f2,…,fn},{x1,x2,…,xn}] is the trace of the vector derivative of f:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-sedwtu
Out[1]=1

More generally, Div[f,x] is the contraction of the last two dimensions of the vector derivative of f:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-odc0g2
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0p2n-h6grl
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-9et684
Out[4]=4

Curl[f,x] is times the HodgeDual of the vector derivative of f, where r is the rank of f:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-gv6u4m
In[2]:=2
✖
https://wolfram.com/xid/0p2n-gxg438
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0p2n-51qwzr
Out[3]=3

For scalar f, Laplacian[f,{x1,x2,…,xn}] is the trace of the second vector derivative of f:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-kl2fgx
Out[1]=1

More generally, Laplacian[f,x] is the contraction of the last two dimensions of the second vector derivative of f:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-4hrf73
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0p2n-gj0fov
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-gmk3he
Out[4]=4

Compute the derivative of Total[a] with respect to a using symbolic arrays:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-f77zxm
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-cg09k2
Out[2]=2

Compare with the results obtained using indexed differentiation:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-jamv1m
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0p2n-hltfmt
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0p2n-g0eji4
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0p2n-hfqdgh
Out[6]=6

ArcCurvature can be defined in terms of D:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-fmkyuh
In[2]:=2
✖
https://wolfram.com/xid/0p2n-ci7ivu
In[3]:=3
✖
https://wolfram.com/xid/0p2n-gfw6nz
Out[3]=3

Systems of differential equations involving D can be solved with DSolve:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-c7nghj
Out[1]=1

Use D to specify a heat equation with homogeneous Dirichlet boundary conditions:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-mzmdbz

The eigensystem for this differential system can be found with DEigensystem:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-idg284

Eigenvalues:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-h2eas0
Out[3]=3

Eigenfunctions:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-zg2obo
Out[4]=4

D can be defined using DifferenceDelta:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-im5ulz
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-sxvvx5
Out[2]=2

D can be defined using DiscreteShift:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-8dmfw6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-k8ubuv
Out[2]=2

The right one-sided derivative is computed with a right-hand limit:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-2xgj8
Out[1]=1

The left one-sided derivative is computed with a left-hand limit:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-cfb5um
Out[2]=2

Note that this function is not differentiable at x==0:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-m1ktz7
Out[3]=3

D assumes that other variables are independent of the differentiation variable:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-mi2wkt
In[2]:=2
✖
https://wolfram.com/xid/0p2n-gqaspq
Out[2]=2

Dt assumes that other variables may depend on the differentiation variable:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-2y7mt7
Out[3]=3

By manually specifying all other variables as constant, Dt can yield the same result as D:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-ouhzor
Out[4]=4

Compute the derivative of an implicit function using D and Solve:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-dper08
In[2]:=2
✖
https://wolfram.com/xid/0p2n-d5tfl4
Out[2]=2

Use ImplicitD to compute the derivative of an implicit function:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-bn0wo1
Out[3]=3

Possible Issues  (5)Common pitfalls and unexpected behavior

Results may not immediately be given in the simplest possible form:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-vb9
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-vgw
Out[2]=2

Functions given in different forms can yield the same derivatives:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-b4k
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-h43
Out[2]=2

D returns generic results that may not account for discontinuities, cusps or other special points:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-lyep5j
In[2]:=2
✖
https://wolfram.com/xid/0p2n-cq0q7t
Out[2]=2

Neither f nor g is differentiable at 0:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-6i4ml
Out[3]=3

f is discontinuous, and g has a cusp:

In[4]:=4
✖
https://wolfram.com/xid/0p2n-479i8
Out[4]=4

If a function can be expanded into a Piecewise expression, D will provide more accurate results:

In[5]:=5
✖
https://wolfram.com/xid/0p2n-gcu45i
Out[5]=5

Cached values for D may miss changes in underlying definitions:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-eq61w0
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0p2n-db1bsn
Out[2]=2

The issue can be resolved by clearing the system cache:

In[3]:=3
✖
https://wolfram.com/xid/0p2n-cx06ag
In[4]:=4
✖
https://wolfram.com/xid/0p2n-2hf3w
Out[4]=4

The variable of differentiation is treated literally:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-jq3j0
Out[1]=1

The following mathematically equivalent input gives 0 because there is no Sin[x] in the first argument:

In[2]:=2
✖
https://wolfram.com/xid/0p2n-twu26u
Out[2]=2

Interactive Examples  (2)Examples with interactive outputs

Find the tangent line to a function:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-ff39fs
In[2]:=2
✖
https://wolfram.com/xid/0p2n-epu042
In[3]:=3
✖
https://wolfram.com/xid/0p2n-blxvh
Out[3]=3

Visualize the secant converging to the tangent as for different base points :

In[1]:=1
✖
https://wolfram.com/xid/0p2n-eoc4e6
In[2]:=2
✖
https://wolfram.com/xid/0p2n-oenv5c
In[3]:=3
✖
https://wolfram.com/xid/0p2n-1xoxke
In[4]:=4
✖
https://wolfram.com/xid/0p2n-l0jtyk
Out[4]=4

Neat Examples  (2)Surprising or curious use cases

Compute the tangent and normal vectors of a 3D parametric function:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-kaeb4w
In[2]:=2
✖
https://wolfram.com/xid/0p2n-cghhqk
In[4]:=4
✖
https://wolfram.com/xid/0p2n-cp8lsa
Out[4]=4

Create a table of n^(th) derivatives:

In[1]:=1
✖
https://wolfram.com/xid/0p2n-chq0g5
In[2]:=2
✖
https://wolfram.com/xid/0p2n-70w5xz
Wolfram Research (1988), D, Wolfram Language function, https://reference.wolfram.com/language/ref/D.html (updated 2024).
✖
Wolfram Research (1988), D, Wolfram Language function, https://reference.wolfram.com/language/ref/D.html (updated 2024).

Text

Wolfram Research (1988), D, Wolfram Language function, https://reference.wolfram.com/language/ref/D.html (updated 2024).

✖
Wolfram Research (1988), D, Wolfram Language function, https://reference.wolfram.com/language/ref/D.html (updated 2024).

CMS

Wolfram Language. 1988. "D." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/D.html.

✖
Wolfram Language. 1988. "D." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/D.html.

APA

Wolfram Language. (1988). D. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/D.html

✖
Wolfram Language. (1988). D. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/D.html

BibTeX

@misc{reference.wolfram_2025_d, author="Wolfram Research", title="{D}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/D.html}", note=[Accessed: 20-June-2025 ]}

✖
@misc{reference.wolfram_2025_d, author="Wolfram Research", title="{D}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/D.html}", note=[Accessed: 20-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_d, organization={Wolfram Research}, title={D}, year={2024}, url={https://reference.wolfram.com/language/ref/D.html}, note=[Accessed: 20-June-2025 ]}

✖
@online{reference.wolfram_2025_d, organization={Wolfram Research}, title={D}, year={2024}, url={https://reference.wolfram.com/language/ref/D.html}, note=[Accessed: 20-June-2025 ]}