Chapter 3 | Derivatives
311
Problem-Solving Strategy: Implicit Differentiation To perform implicit differentiation on an equation that defines a function y implicitly in terms of a variable x , use the following steps: 1. Take the derivative of both sides of the equation. Keep in mind that y is a function of x . Consequently, whereas d dx (sin x ) =cos x , d dx (sin y ) =cos y dy dx because we must use the chain rule to differentiate sin y with respect
to x .
2. Rewrite the equation so that all terms containing dy dx
are on the left and all terms that do not contain dy dx are
on the right.
dy dx
3. Factor out
on the left.
dy dx by dividing both sides of the equation by an appropriate algebraic expression.
4. Solve for
Example 3.68 Using Implicit Differentiation
Assuming that y is defined implicitly by the equation x 2 + y 2 =25, find dy dx .
Solution Follow the steps in the problem-solving strategy. d dx ⎛ ⎝ x 2 + y 2 ⎞ ⎠ = d
dx (25) Step 1. Differentiate both sides of the equation.
Step 1.1. Use the sum rule on the left. On the right d dx (25) =0. Step 1.2. Take the derivatives, so d dx ⎛ ⎝ x 2 ⎞
⎛ ⎝ x 2
⎞ ⎠ + d
⎛ ⎝ y 2
⎞ ⎠ = 0
d dx
dx
⎠ =2 x
dy dx = 0
2 x +2 y
dy dx .
⎛ ⎝ y 2
⎞ ⎠ =2 y
and d
dx
Step 2. Keep the terms with dy dx on the left. Move the remaining terms to the right. Step 4. Divide both sides of the equation by 2 y .(Step 3 does not apply in this case.)
dy dx = −2
2 y
x
dy dx = −
x y
Analysis Note that the resulting expression for
dy dx is in terms of both the independent variable x and the dependent
variable y . Although in some cases it may be possible to express dy dx
in terms of x only, it is generally not
possible to do so.
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