Chapter 3 | Derivatives
263
3.3 EXERCISES For the following exercises, find f ′( x ) for each function. 106. f ( x ) = x 7 +10 107. f ( x ) =5 x 3 − x +1 108. f ( x ) =4 x 2 −7 x 109. f ( x ) =8 x 4 +9 x 2 −1 110. f ( x ) = x 4 + 2 x
g ( x ) 7
122. h ( x ) =4 f ( x )+
123. h ( x ) = x 3 f ( x )
f ( x ) g ( x ) 2 f ( x ) g ( x )+2
124. h ( x ) =
125. h ( x ) = 3
For the following exercises, assume that f ( x ) and g ( x ) are both differentiable functions with values as given in the following table. Use the following table to calculate the following derivatives.
111. f ( x ) =3 x ⎛
⎞ ⎠
⎝ 18 x 4 + 13
x +1
112. f ( x ) = ( x +2) ⎛
⎝ 2 x 2 −3 ⎞ ⎠
1
2 3
4
x
⎛ ⎝ 2
⎞ ⎠
f ( x )
3
5 −2 0
113. f ( x ) = x 2
+ 5 x 3
x 2
g ( x )
2
3 −4 6
3 +2 x 2 −4 3
114. f ( x ) = x
f ′( x )
−1 7 8
−3
3 −2 x +1 x 2
115. f ( x ) = 4 x
g ′( x )
4
1 2
9
2 +4 x 2 −4
116. f ( x ) = x
x +9 x 2 −7 x +1
117. f ( x ) =
126. Find h ′(1) if h ( x ) = xf ( x )+4 g ( x ).
127. Find h ′(2) if h ( x ) = f ( x ) g ( x ) . 128. Find h ′(3) if h ( x ) =2 x + f ( x ) g ( x ).
For the following exercises, find the equation of the tangent line T ( x ) to the graph of the given function at the indicated point. Use a graphing calculator to graph the function and the tangent line. 118. [T] y =3 x 2 +4 x +1 at (0, 1) 119. [T] y =2 x +1 at (4, 5) 120. [T] y = 2 x x −1 at (−1, 1) 121. [T] y = 2 x − 3 x 2 at (1, −1) For the following exercises, assume that f ( x ) and g ( x ) are both differentiable functions for all x . Find the derivative of each of the functions h ( x ).
g ( x ) f ( x ) .
129. Find h ′(4) if h ( x ) = 1 x +
For the following exercises, use the following figure to find the indicated derivatives, if they exist.
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