326
Chapter 3 | Derivatives
Find the derivative of f ( x ) = ln ⎛
⎝ x 3 +3 x −4 ⎞ ⎠ .
Solution Use Equation 3.31 directly. f ′( x ) = 1
⎛ ⎝ 3 x 2 +3 ⎞ ⎠
Use g ( x ) = x 3 +3 x −4 in h ′( x ) = 1 g ( x )
g ′( x ).
·
x 3 +3 x −4 2 +3 x 3 +3 x −4
= 3 x
Rewrite.
Example 3.78 Using Properties of Logarithms in a Derivative
Find the derivative of f ( x ) = ln ⎛
⎞ ⎠ .
⎝ x 2 sin x 2 x +1
Solution At first glance, taking this derivative appears rather complicated. However, by using the properties of logarithms prior to finding the derivative, we can make the problem much simpler. f ( x ) = ln ⎛ ⎝ x 2 sin x 2 x +1 ⎞ ⎠ =2ln x +ln(sin x )−ln(2 x +1) Apply properties of logarithms. f ′( x ) = 2 x +cot x − 2 2 x +1 Apply sum rule and h ′( x ) = 1 g ( x ) g ′( x ).
3.52
Differentiate: f ( x ) = ln(3 x +2) 5 .
Now that we can differentiate the natural logarithmic function, we can use this result to find the derivatives of y = log b x and y = b x for b >0, b ≠1.
Theorem 3.16: Derivatives of General Exponential and Logarithmic Functions Let b >0, b ≠1, and let g ( x ) be a differentiable function. i. If, y = log b x , then
dy dx = 1 x
(3.32)
.
ln b
More generally, if h ( x ) = log b ⎛
⎞ ⎠ , then for all values of x for which g ( x ) >0,
⎝ g ( x )
g ′( x ) g ( x )ln b .
(3.33)
h ′( x ) =
ii. If y = b x , then
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