Question

1. (0 / 0.25 points)

find the derivative of the function.
$f(x) = \ln(9x)$
$f(x) = \frac{1}{9x}$
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1. (0 / 0.25 points)

find the derivative of the function.
$y = (\ln(x))^6$
$y = 6(\ln(x))^5 \cdot (\frac{1}{x})^6$

Explanation:

Step1: Differentiate $f(x)=\ln(9x)$

First, use logarithm property: $\ln(9x)=\ln9+\ln x$.
Derivative of constant $\ln9$ is 0, derivative of $\ln x$ is $\frac{1}{x}$.
Or use chain rule: Let $u=9x$, $f(u)=\ln u$.
$f'(x)=\frac{1}{u}\cdot u'=\frac{1}{9x}\cdot9=\frac{1}{x}$

Step2: Differentiate $y=(\ln x)^6$

Use chain rule: Let $u=\ln x$, $y=u^6$.
$y'=6u^5\cdot u'=6(\ln x)^5\cdot\frac{1}{x}$

Answer:

For $f(x)=\ln(9x)$: $f'(x)=\frac{1}{x}$
For $y=(\ln x)^6$: $y'=\frac{6(\ln x)^5}{x}$