Calculus

Differentiation of Functions

Differentiation Logo

Implicit Differentiation

If a function is described by the equation y = f (x) where the variable y is on the left side, and the right side depends only on the independent variable x, then the function is said to be given explicitly. For example, the following functions are defined explicitly:

In many problems, however, the function can be defined in implicit form, that is by the equation

Of course, any explicit function can be written in an implicit form. So the above functions can be represented as

The inverse transformation cannot be always performed. There are often functions defined by an implicit equation that cannot be resolved with respect to the variable Examples of such implicit functions are

The good news is that we do not need to convert an implicitly defined function into an explicit form to find the derivative If is defined implicitly as a function of by an equation we proceed as follows:

  1. Differentiate both sides of the equation with respect to , assuming that is a differentiable function of and using the chain rule. The derivative of zero (in the right side) will also be equal to zero.

    Note: If the right side is different from zero, that is the implicit equation has the form
    then we differentiate the left and right side of the equation.
  2. Solve the resulting equation for the derivative .

In the examples below find the derivative of the implicit function.

Solved Problems

Example 1.

Find the derivative of the function given by the equation where is a parameter.

Solution.

This equation is the canonical equation of a parabola. Differentiating the left and right sides with respect to , we have:

Example 2.

Differentiate implicitly the function given by the equation

Solution.

Differentiate both sides with respect to

which results in

Example 3.

Calculate the derivative at the point of the function given by the equation

Solution.

We differentiate both sides of the equation with respect to and solve for

Substitute the coordinates

Example 4.

Find the equation of the tangent line to the curve at the point

Solution.

Differentiate both sides of the equation with respect to

Then . At the point we have Hence, the equation of the tangent line is given by

Example 5.

Calculate the derivative of the function given by the equation under condition

Solution.

We differentiate both sides of the equation implicitly with respect to (we consider the left side as a composite function and use the chain rule):

When the original equation becomes

Substituting the values and , we obtain:

It follows from here that at

Example 6.

Given the equation of a circle of radius centered at the origin. Find the derivative

Solution.

Differentiate both sides of the equation with respect to

In this case, we can solve for directly from the equation for the upper half-circle: So we get

Example 7.

Solution.

We take the derivative of each term treating as a function of

Solve this equation for

Example 8.

Solution.

We differentiate the left and right sides of the equation with respect to considering as a composite function of

From this relation we find

This derivative exists provided

Example 9.

Solution.

Differentiate both sides term-by-term with respect to

Solve this equation for

Example 10.

Calculate the derivative at the point of the function given by the equation

Solution.

We differentiate this equation with respect to and solve for

Substitute the coordinates

See more problems on Page 2.

Page 1 Page 2