730
Chapter 6 | Applications of Integration
dy dx =
d dx
⎛ ⎝ log a x ⎞ ⎠
⎛ ⎝ ln x ln a
⎞ ⎠
= d
dx ⎛ ⎝ 1
⎞ ⎠ d
dx (ln
x )
=
ln a
= 1
· 1 x
ln a
= 1 x
.
ln a
Theorem 6.21: Derivatives of General Logarithm Functions Let a >0. Then, d dx log a x = 1 x ln a .
Example 6.40 Calculating Derivatives of General Exponential and Logarithm Functions
Evaluate the following derivatives: a. d dt ⎛ ⎝ 4 t ·2 t 2 ⎞ ⎠
⎛ ⎝ 7 x 2 +4 ⎞ ⎠
d dx log 8
b.
Solution We need to apply the chain rule as necessary. a. d dt ⎛ ⎝ 4 t ·2 t 2 ⎞ ⎠ = d dt ⎛ ⎝ 2 2 t ·2 t 2 ⎞ ⎠ = d dt ⎛ ⎝ 2 2 t + t 2 ⎞
2
⎠ =2 2 t + t
ln(2)(2+2 t )
⎛ ⎝ 7 x 2 +4 ⎞
d dx log 8
(14 x )
⎠ = 1 ⎛ ⎝ 7 x 2 +4 ⎞
b.
⎠ (ln8)
6.40
Evaluate the following derivatives:
t 4
a. d
dt 4
⎛ ⎝ x 2 +1
⎞ ⎠
d dx log 3
b.
Example 6.41 Integrating General Exponential Functions
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