Learning Objectives
- Find the derivative of a complicated function by using implicit differentiation.
- Use implicit differentiation to determine the equation of a tangent line.
We have already studied how to find equations of tangent lines to functions and the rate of change of a function at a specific point. In all these cases we had the explicit equation for the function and differentiated these functions explicitly. Suppose instead that we want to determine the equation of a tangent line to an arbitrary curve or the rate of change of an arbitrary curve at a point. In this section, we solve these problems by finding the derivatives of functions that define
Implicit Differentiation
In most discussions of math, if the dependent variable
Implicit differentiation allows us to find slopes of tangents to curves that are clearly not functions (they fail the vertical line test). We are using the idea that portions of
In general, an equation defines a function implicitly if the function satisfies that equation. An equation may define many different functions implicitly. For example, the functions

If we want to find the slope of the line tangent to the graph of
Problem-Solving Strategy: Implicit Differentiation
To perform implicit differentiation on an equation that defines a function
- Take the derivative of both sides of the equation. Keep in mind that
is a function of . Consequently, whereas because we must use the Chain Rule to differentiate with respect to . - Rewrite the equation so that all terms containing
are on the left and all terms that do not contain are on the right. - Factor out
on the left. - Solve for
by dividing both sides of the equation by an appropriate algebraic expression.
Using Implicit Differentiation
Assuming that
Solution
Follow the steps in the problem-solving strategy.
Using Implicit Differentiation and the Product Rule
Assuming that
Solution
Using Implicit Differentiation to Find a Second Derivative
Find
Solution
In (Figure) , we showed that
At this point we have found an expression for
Find
Hint
Follow the problem solving strategy, remembering to apply the chain rule to differentiate
Solution
Finding Tangent Lines Implicitly
Now that we have seen the technique of implicit differentiation, we can apply it to the problem of finding equations of tangent lines to curves described by equations.
Finding a Tangent Line to a Circle
Find the equation of the line tangent to the curve
Solution
Although we could find this equation without using implicit differentiation, using that method makes it much easier. In (Figure) , we found
The slope of the tangent line is found by substituting
Using the point

Finding the Equation of the Tangent Line to a Curve
Find the equation of the line tangent to the graph of

Solution
Begin by finding
Next, substitute
Finally, substitute into the point-slope equation of the line and solve for
Applying Implicit Differentiation
In a simple video game, a rocket travels in an elliptical orbit whose path is described by the equation
Solution
To solve this problem, we must determine where the line tangent to the graph of
Differentiating, we have
Solving for
The slope of the tangent line is
Find the equation of the line tangent to the hyperbola
Hint
Using implicit differentiation, you should find that
Solution
Key Concepts
- We use implicit differentiation to find derivatives of implicitly defined functions (functions defined by equations).
- By using implicit differentiation, we can find the equation of a tangent line to the graph of a curve.
For the following exercises, use implicit differentiation to find
1.
2.
Solution
3.
4.
Solution
5.
6.
Solution
7.
8.
Solution
9.
10.
11.
12.
For the following exercises, find
13.
14.
15.
16.
For the following exercises, find the equation of the tangent line to the graph of the given equation at the indicated point. Use a calculator or computer software to graph the function and the tangent line.
17. [T]
18. [T]
Solution
19. [T]
20. [T]
Solution
21. [T]
22. [T]
Solution
23. [T] The graph of a folium of Descartes with equation
- Find the equation of the tangent line at the point
. Graph the tangent line along with the folium. - Find the equation of the normal line to the tangent line in a. at the point
.
24. For the equation
- Find the equation of the normal to the tangent line at the point
. - At what other point does the normal line in a. intersect the graph of the equation?
Solution
a.
b.
25. Find all points on the graph of
26. For the equation
- Find the
-intercept(s). - Find the slope of the tangent line(s) at the
-intercept(s). - What does the value(s) in b. indicate about the tangent line(s)?
Solution
a.
b. Slope is -2 at both intercepts
c. They are parallel since the slope is the same at both intercepts.
27. Find the equation of the tangent line to the graph of the equation
28. Find the equation of the tangent line to the graph of the equation
Solution
29. Find
30. [T] The number of cell phones produced when
- Find
and evaluate at the point . - Interpret the result of a.
Solution
a. -0.5926
b. When $81 is spent on labor and $16 is spent on capital, the amount spent on capital is decreasing by $0.5926 per $1 spent on labor.
31. [T] The number of cars produced when
(Both
- Find
and evaluate at the point . - Interpret the result of a.
32. The volume of a right circular cone of radius
Solution
For the following exercises, consider a closed rectangular box with a square base with side
33. Find an equation for the surface area of the rectangular box,
34. If the surface area of the rectangular box is 78 square feet, find
Solution
For the following exercises, use implicit differentiation to determine
35.
36.
Solution
37.
Glossary
- implicit differentiation
- is a technique for computing
for a function defined by an equation, accomplished by differentiating both sides of the equation (remembering to treat the variable as a function) and solving for
Analysis
Note that the resulting expression for is in terms of both the independent variable and the dependent variable . Although in some cases it may be possible to express in terms of only, it is generally not possible to do so.