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Chapter 3 | Derivatives
Solution First recall that sin 3 x = (sin x ) 3 , so we can rewrite h ( x ) = sin 3 x as h ( x ) = (sin x ) 3 . Applying the power rule with g ( x ) = sin x , we obtain h ′( x ) =3(sin x ) 2 cos x =3sin 2 x cos x .
Example 3.50 Finding the Equation of a Tangent Line
Find the equation of a line tangent to the graph of h ( x ) = 1 (3 x −5) 2
at x =2.
Solution Because we are finding an equation of a line, we need a point. The x -coordinate of the point is 2. To find the y -coordinate, substitute 2 into h ( x ). Since h (2) = 1 ⎛ ⎝ 3(2)−5 ⎞ ⎠ 2 =1, the point is (2, 1). For the slope, we need h ′(2). To find h ′( x ), first we rewrite h ( x ) = (3 x −5) −2 and apply the power rule to obtain h ′( x ) =−2(3 x −5) −3 (3) =−6(3 x −5) −3 . By substituting, we have h ′(2) =−6 ⎛ ⎝ 3(2)−5 ⎞ ⎠ −3 =−6. Therefore, the line has equation y −1=−6( x −2). Rewriting, the equation of the line is y =−6 x +13.
3
3.35
Find the equation of the line tangent to the graph of f ( x ) = ⎛
⎞ ⎠
⎝ x 2 −2
at x =−2.
Combining the Chain Rule with Other Rules Now that we can combine the chain rule and the power rule, we examine how to combine the chain rule with the other rules we have learned. In particular, we can use it with the formulas for the derivatives of trigonometric functions or with the product rule. Example 3.51 Using the Chain Rule on a General Cosine Function
Find the derivative of h ( x ) =cos ⎛
⎞ ⎠ .
⎝ g ( x )
Solution
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