Chapter 3 | Derivatives
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3.8 | Implicit Differentiation
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 y implicitly in terms of x . Implicit Differentiation In most discussions of math, if the dependent variable y is a function of the independent variable x , we express y in terms of x . If this is the case, we say that y is an explicit function of x . For example, when we write the equation y = x 2 +1, we are defining y explicitly in terms of x . On the other hand, if the relationship between the function y and the variable x is expressed by an equation where y is not expressed entirely in terms of x , we say that the equation defines y implicitly in terms of x . For example, the equation y − x 2 =1 defines the function y = x 2 +1 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 y are functions that satisfy the given equation, but that y is not actually a function of x . 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 y = 25− x 2 and y = , which are illustrated in Figure 3.30 , are just three of the many
⎧ ⎩
⎨ 25− x 2 if −5< x <0 − 25− x 2 if 0< x <25
functions defined implicitly by the equation x 2 + y 2 =25.
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