Chapter 6 | Applications of Integration
723
Figure 6.76 The graph of f ( x ) = ln x shows that it is a continuous function.
Example 6.35 Calculating Derivatives of Natural Logarithms
Calculate the following derivatives: a. d dx ln ⎛ ⎝ 5 x 3 −2 ⎞ ⎠
d dx
⎠ 2
⎛ ⎝ ln(3 x ) ⎞
b.
Solution We need to apply the chain rule in both cases.
2 5 x 3 −2
⎛ ⎝ 5 x 3 −2 ⎞
⎠ = 15 x
d dx ln
a.
⎠ 2 = 2 ⎛
⎝ ln(3 x ) ⎞
= 2 ⎛
⎝ ln(3 x ) ⎞ ⎠ x
⎠ ·3
d dx
⎛ ⎝ ln(3 x ) ⎞
b.
3 x
6.35
Calculate the following derivatives:
⎛ ⎝ 2 x 2 + x ⎞ ⎠
d dx ln
a.
2
⎛ ⎝ ln
⎛ ⎝ x 3
⎞ ⎠
⎞ ⎠
d dx
b.
Note that if we use the absolute value function and create a new function ln | x |, we can extend the domain of the natural logarithm to include x <0. Then ⎛ ⎝ d /( dx ) ⎞ ⎠ ln | x | =1/ x . This gives rise to the familiar integration formula.
Theorem 6.17: Integral of (1/ u ) du The natural logarithm is the antiderivative of the function f ( u ) =1/ u :
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