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Chapter 3 | Derivatives
k ′( x ) = d dx ⎛
⎝ 3 h ( x )+ x 2 g ( x ) ⎞
⎛ ⎝ x 2 g ( x )
⎞ ⎠
⎠ = d
⎠ + d
⎛ ⎝ 3 h ( x ) ⎞
Apply the sum rule.
dx
dx
Apply the constant multiple rule to differentiate 3 h ( x )and the product rule to differentiate x 2 g ( x ).
⎛ ⎝ d
⎞ ⎠
⎛ ⎝ x 2
⎞ ⎠ g ( x )+ d dx ⎛
=3 d
⎞ ⎠ x 2
⎛ ⎝ h ( x )
⎞ ⎠ +
⎝ g ( x )
dx
dx
=3 h ′( x )+2 xg ( x )+ g ′( x ) x 2
Example 3.29 Extending the Product Rule
For k ( x ) = f ( x ) g ( x ) h ( x ), express k ′( x ) in terms of f ( x ), g ( x ), h ( x ), and their derivatives.
Solution We can think of the function k ( x ) as the product of the function f ( x ) g ( x ) and the function h ( x ). That is, k ( x ) = ⎛ ⎝ f ( x ) g ( x ) ⎞ ⎠ · h ( x ). Thus, k ′( x ) = d dx ⎛ ⎝ f ( x ) g ( x ) ⎞ ⎠ · h ( x )+ d dx ⎛ ⎝ h ( x ) ⎞ ⎠ · ⎛ ⎝ f ( x ) g ( x ) ⎞ ⎠
Apply the product rule to the product of f ( x ) g ( x ) and h ( x ). Apply the product rule to f ( x ) g ( x ).
= ⎛ ⎝ f ′( x ) g ( x )+ g ′( x ) f ( x )) h ( x )+ h ′( x ) f ( x ) g ( x ) = f ′( x ) g ( x ) h ( x )+ f ( x ) g ′( x ) h ( x )+ f ( x ) g ( x ) h ′( x ).
Simplify.
Example 3.30 Combining the Quotient Rule and the Product Rule
x 3 k ( x ) 3 x +2 ,
For h ( x ) = 2
find h ′( x ).
Solution This procedure is typical for finding the derivative of a rational function.
⎛ ⎝ 2 x 3 k ( x ) ⎞
⎛ ⎝ 2 x 3 k ( x ) ⎞ ⎠
d dx
⎠ · (3 x +2)− d
dx (3 x +2) ·
h ′( x ) =
Apply the quotient rule.
(3 x +2) 2
⎛ ⎝ 6 x 2 k ( x )+ k ′( x ) ·2 x 3 ⎞
⎠ (3 x +2)−3 ⎛
⎝ 2 x 3 k ( x ) ⎞ ⎠
Apply the product rule to find d dx ⎛ ⎝ 2 x 3 k ( x ) ⎞ ⎠ .Use d dx (3
=
(3 x +2) 2
x +2) =3.
3 k ( x )+18 x 3 k ( x )+12 x 2 k ( x )+6 x 4 k ′( x )+4 x 3 k ′( x ) (3 x +2) 2
= −6 x
Simplify.
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