ImplicitD

ImplicitD[eqn,y,x]

gives the partial derivative , assuming that the variable y represents an implicit function defined by the equation eqn.

ImplicitD[f,eqn,y,x]

gives the partial derivative , assuming that the variable y represents an implicit function defined by the equation eqn.

ImplicitD[f,{eqn1,,eqnk},{y1,,yk},x]

gives the partial derivative , assuming that the variables y1,,yk represent implicit functions defined by the system of equations eqn1eqnk.

ImplicitD[f,eqns,ys,{x,n}]

gives the multiple derivative .

ImplicitD[f,eqns,ys,x1,x2,]

gives the partial derivative .

ImplicitD[f,eqns,ys,{x1,n1},{x2,n2},]

gives the multiple partial derivative .

ImplicitD[f,eqns,ys,{{x1,x2,}}]

for a scalar f gives the vector derivative .

ImplicitD[f,eqns,ys,{array}]

gives an array derivative.

Details

  • ImplicitD is typically used to compute derivatives of implicitly defined functions.
  • If variables x and y satisfy an equation , then, under certain conditions spelled out in the following, y can be locally treated as a function of x, and the derivative of this function can be expressed in terms of partial derivatives of g.
  • If a function is continuously differentiable, and , then the implicit function theorem guarantees that in a neighborhood of there is a unique function such that and . is called an implicit function defined by the equation . Thus, .
  • ImplicitD[f,g==0,y,] assumes that is continuously differentiable and requires that .
  • Similarly, if variables and satisfy a system of equations then, under certain conditions spelled out in the following, can be locally treated as functions of , and the derivatives of these functions can be expressed in terms of partial derivatives of .
  • If functions are continuously differentiable, and the Jacobian matrix is invertible, then the implicit function theorem guarantees that in a neighborhood of , there are unique functions such that and . Functions are called implicit functions defined by the equations . Thus, .
  • ImplicitD[f,{g1==0,,gk==0},{y1,,yk},] assumes that are continuously differentiable and requires that the Jacobian matrix is invertible.
  • For lists, ImplicitD[{f1,f2,},] is equivalent to {ImplicitD[f1,],ImplicitD[f2,],}, recursively.
  • ImplicitD[eqns,ys,], where eqns is an equation or a list of equations, is equivalent to ImplicitD[ys,eqns,ys,].
  • ImplicitD[f,eqns,ys,{array}] effectively threads ImplicitD over each element of array.
  • All expressions that do not explicitly depend on the differentiation variable or on the variables representing implicit functions are taken to have zero partial derivative.

Examples

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Basic Examples  (5)

Derivative of with respect to , where is constrained by the equation :

Second derivative of , assuming that :

Derivative involving two implicitly defined functions:

Derivative with respect to and :

Derivative involving symbolic functions and :

Scope  (9)

Derivative of an implicitly defined function:

Derivative of an expression involving an implicit function defined by a polynomial equation:

Derivative of an expression involving an implicit function defined by a transcendental equation:

Derivative of an expression involving two implicit functions defined by a pair of equations:

Third derivative of an expression involving an implicit function:

The mixed partial derivative of an expression involving an implicit function:

The mixed partial derivative :

Gradient of an expression involving an implicit function:

Jacobian of an expression involving an implicit function:

Equation defining an implicit function needs to have a nonzero derivative with respect to :

Equations defining implicit functions need to have an invertible Jacobian with respect to :

Applications  (3)

Compute the slope of the tangent line to a curve:

The slope of the tangent line is equal to the derivative of with respect to :

Show tangent lines at six points on the curve:

Find tangent planes to a surface:

Compute the gradient of with respect to and :

Show tangent planes at three points on the surface:

Verify an implicit solution to a differential equation:

The derivative of the solution is equal to the right-hand side of the differential equation:

Properties & Relations  (4)

Equation defines an implicit function in a neighborhood of any point where :

The derivative of the implicit function equals :

The derivative has singularities at points where :

Compute the derivative of an implicit function using D:

Compare with the result obtained using ImplicitD:

Use SolveValues to find an explicit solution of :

Compare the derivative of the solution with the result obtained using ImplicitD:

Root[g,k] represents a solution of g[y]:

Compare the derivative of with the result obtained using ImplicitD:

Possible Issues  (1)

ImplicitD[f,{y,g},] requires that :

The result is valid when and is singular otherwise:

Wolfram Research (2022), ImplicitD, Wolfram Language function, https://reference.wolfram.com/language/ref/ImplicitD.html.

Text

Wolfram Research (2022), ImplicitD, Wolfram Language function, https://reference.wolfram.com/language/ref/ImplicitD.html.

CMS

Wolfram Language. 2022. "ImplicitD." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ImplicitD.html.

APA

Wolfram Language. (2022). ImplicitD. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ImplicitD.html

BibTeX

@misc{reference.wolfram_2022_implicitd, author="Wolfram Research", title="{ImplicitD}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/ImplicitD.html}", note=[Accessed: 08-August-2022 ]}

BibLaTeX

@online{reference.wolfram_2022_implicitd, organization={Wolfram Research}, title={ImplicitD}, year={2022}, url={https://reference.wolfram.com/language/ref/ImplicitD.html}, note=[Accessed: 08-August-2022 ]}