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,…,g_k==0},{y₁,…,y_k},…] 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

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: