Chapter 3 | Derivatives
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Figure 3.12 The derivative f ′( x ) is positive everywhere because the function f ( x ) is increasing.
In Example 3.12 we found that for f ( x ) = x 2 −2 x , f ′( x ) =2 x −2. The graphs of these functions are shown in Figure 3.13 . Observe that f ( x ) is decreasing for x <1. For these same values of x , f ′( x ) <0. For values of x >1, f ( x ) is increasing and f ′( x ) >0. Also, f ( x ) has a horizontal tangent at x =1 and f ′(1) =0.
Figure 3.13 The derivative f ′( x ) <0 where the function f ( x ) is decreasing and f ′( x ) >0 where f ( x ) is increasing. The derivative is zero where the function has a horizontal tangent.
Example 3.13 Sketching a Derivative Using a Function
Use the following graph of f ( x ) to sketch a graph of f ′( x ).
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