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Chapter 3 | Derivatives
Figure 3.30 The equation x 2 + y 2 =25 defines many functions implicitly.
If we want to find the slope of the line tangent to the graph of x 2 + y 2 =25 at the point (3, 4), we could evaluate the derivative of the function y = 25− x 2 at x =3. On the other hand, if we want the slope of the tangent line at the point (3, −4), we could use the derivative of y =− 25− x 2 . 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
dy dx
differentiation is described in the following problem-solving strategy.
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