Chapter 3 | Derivatives
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□ Figure 3.27 shows the relationship between the graph of f ( x ) = sin x and its derivative f ′( x ) =cos x . Notice that at the points where f ( x ) = sin x has a horizontal tangent, its derivative f ′( x ) =cos x takes on the value zero. We also see that where f ( x ) = sin x is increasing, f ′( x ) =cos x >0 and where f ( x ) = sin x is decreasing, f ′( x ) =cos x <0.
Figure 3.27 Where f ( x ) has a maximum or a minimum, f ′( x ) =0 that is, f ′( x ) =0 where f ( x ) has a horizontal tangent. These points are noted with dots on the graphs.
Example 3.39 Differentiating a Function Containing sin x
Find the derivative of f ( x ) =5 x 3 sin x .
Solution Using the product rule, we have
f ′( x ) = d dx ⎛
⎞ ⎠ · sin x + d
⎝ 5 x 3
x ) ·5 x 3
dx (sin
=15 x 2 · sin x +cos x ·5 x 3 .
After simplifying, we obtain
f ′( x ) =15 x 2 sin x +5 x 3 cos x .
Find the derivative of f ( x ) = sin x cos x .
3.25
Example 3.40 Finding the Derivative of a Function Containing cos x
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