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 eqn1∧…∧eqnk.
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
![](Files/ImplicitD.en/details_1.png)
![](Files/ImplicitD.en/details_2.png)
![](Files/ImplicitD.en/details_3.png)
- 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.
![](Files/ImplicitD.en/Image_9.gif)
![](Files/ImplicitD.en/Image_30.gif)
Examples
open allclose allBasic Examples (5)
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
:
![](Files/ImplicitD.en/57.gif)
Equations defining implicit functions need to have an invertible Jacobian with respect to
:
![](Files/ImplicitD.en/60.gif)
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)
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