Section 3.8 Implicit Differentiation
Learning Objectives.
Find the derivative of a complicated function by using implicit differentiation.
Use implicit differentiation to determine the equation of a tangent line.
Subsection 3.8.1 Implicit Differentiation
In most discussions of math, if the dependent variable
Note 3.139. 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 toRewrite 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.
Example 3.140. Using Implicit Differentiation.
Assuming that
Follow the steps in the problem-solving strategy.
Note 3.141. Analysis.
Note that the resulting expression for
Example 3.142. Using Implicit Differentiation.
Assuming that
Example 3.143. Using Implicit Differentiation to Find a Second Derivative.
Find
In Example 3.140, we showed that
At this point we have found an expression for
Checkpoint 3.144.
Find
Subsection 3.8.2 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.Example 3.145. Finding a Tangent Line to a Circle.
Find the equation of the line tangent to the curve
Although we could find this equation without using implicit differentiation, using that method makes it much easier. In Example 3.140, we found
The slope of the tangent line is found by substituting
Using the point

Example 3.147. Finding the Equation of the Tangent Line to a Curve.
Find the equation of the line tangent to the graph of

Begin by finding
Next, substitute
Finally, substitute into the point-slope equation of the line to obtain
Example 3.149. Applying Implicit Differentiation.
In a simple video game, a rocket travels in an elliptical orbit whose path is described by the equation
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
Checkpoint 3.150.
Find the equation of the line tangent to the hyperbola
Subsection 3.8.3 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.