Learning Objectives
- 3.8.1 Find the derivative of a complicated function by using implicit differentiation.
- 3.8.2 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 implicitly in terms of
Implicit Differentiation
In most discussions of math, if the dependent variable is a function of the independent variable we express y in terms of If this is the case, we say that is an explicit function of For example, when we write the equation we are defining y explicitly in terms of On the other hand, if the relationship between the function and the variable is expressed by an equation where is not expressed entirely in terms of we say that the equation defines y implicitly in terms of For example, the equation defines the function implicitly.
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 are functions that satisfy the given equation, but that is not actually a function 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
and which are illustrated in Figure 3.30, are just three of the many functions defined implicitly by the equation
If we want to find the slope of the line tangent to the graph of at the point we could evaluate the derivative of the function at On the other hand, if we want the slope of the tangent line at the point we could use the derivative of However, it is not always easy to solve for a function defined implicitly by an equation. Fortunately, the technique of implicit differentiation allows us to find the derivative of an implicitly defined function without ever solving for the function explicitly. The process of finding using implicit differentiation is described in the following problem-solving strategy.
To perform implicit differentiation on an equation that defines a function implicitly in terms of a variable use the following steps:
- Take the derivative of both sides of the equation. Keep in mind that y is a function of x. 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.
Example 3.68
Using Implicit Differentiation
Assuming that is defined implicitly by the equation find
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.
Example 3.69
Using Implicit Differentiation and the Product Rule
Assuming that is defined implicitly by the equation find
Example 3.70
Using Implicit Differentiation to Find a Second Derivative
Find if
Find for defined implicitly by the equation
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 3.71
Finding a Tangent Line to a Circle
Find the equation of the line tangent to the curve at the point
Example 3.72
Finding the Equation of the Tangent Line to a Curve
Find the equation of the line tangent to the graph of at the point (Figure 3.32). This curve is known as the folium (or leaf) of Descartes.
Example 3.73
Applying Implicit Differentiation
In a simple video game, a rocket travels in an elliptical orbit whose path is described by the equation The rocket can fire missiles along lines tangent to its path. The object of the game is to destroy an incoming asteroid traveling along the positive x-axis toward If the rocket fires a missile when it is located at where will it intersect the x-axis?
Find the equation of the line tangent to the hyperbola at the point
Section 3.8 Exercises
For the following exercises, use implicit differentiation to find
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.
[T]
[T]
[T]
[T] The graph of a folium of Descartes with equation is given in the following graph.
- 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
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?
Find all points on the graph of at which the tangent line is vertical.
For the equation
- Find the -intercept(s).
- Find the slope of the tangent line(s) at the x-intercept(s).
- What does the value(s) in b. indicate about the tangent line(s)?
Find the equation of the tangent line to the graph of the equation at the point
Find and for
[T] The number of cell phones produced when dollars is spent on labor and dollars is spent on capital invested by a manufacturer can be modeled by the equation
- Find and evaluate at the point
- Interpret the result of a.
[T] The number of cars produced when dollars is spent on labor and dollars is spent on capital invested by a manufacturer can be modeled by the equation
(Both and are measured in thousands of dollars.)
- Find and evaluate at the point
- Interpret the result of a.
The volume of a right circular cone of radius and height is given by Suppose that the volume of the cone is a constant. Find when and
For the following exercises, consider a closed rectangular box with a square base with side and height
Find an equation for the surface area of the rectangular box,
For the following exercises, use implicit differentiation to determine Does the answer agree with the formulas we have previously determined?