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
The good news is that we do not need to convert an implicitly defined function into an explicit form to find the derivative
- 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 formthen we differentiate the left and right side of the equation. - 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
Solution.
This equation is the canonical equation of a parabola. Differentiating the left and right sides with respect to
Example 2.
Differentiate implicitly the function
Solution.
Differentiate both sides with respect to
which results in
Example 3.
Calculate the derivative at the point
Solution.
We differentiate both sides of the equation with respect to
Substitute the coordinates
Example 4.
Find the equation of the tangent line to the curve
Solution.
Differentiate both sides of the equation with respect to
Then
Example 5.
Calculate the derivative of the function
Solution.
We differentiate both sides of the equation implicitly with respect to
When
Substituting the values
It follows from here that
Example 6.
Given the equation of a circle
Solution.
Differentiate both sides of the equation with respect to
In this case, we can solve for
Example 7.
Solution.
We take the derivative of each term treating
Solve this equation for
Example 8.
Solution.
We differentiate the left and right sides of the equation with respect to
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
Solution.
We differentiate this equation with respect to
Substitute the coordinates