# 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: 