# Math 22 Derivatives of Logarithmic Functions

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## Derivative of the Natural Logarithmic Function

 Let ${\displaystyle u}$ be a differentiable function of ${\displaystyle x}$.
1.${\displaystyle {\frac {d}{dx}}[\ln x]={\frac {1}{x}}}$ for ${\displaystyle x>0}$

2.${\displaystyle {\frac {d}{dx}}[\ln u]={\frac {1}{u}}{\frac {du}{dx}}}$ for ${\displaystyle u>0}$


Exercises Find the derivative of the function

a) ${\displaystyle f(x)=\ln(7x)}$

Solution:
${\displaystyle f'(x)={\frac {1}{7x}}(7x)'={\frac {1}{7x}}7={\frac {1}{x}}}$

b) ${\displaystyle f(x)=\ln(x^{8})}$

Solution:
Solution 1: ${\displaystyle f'(x)={\frac {1}{x^{8}}}(x^{8})'={\frac {8x^{7}}{x^{8}}}={\frac {8}{x}}}$
Solution 2: ${\displaystyle f(x)=\ln(x^{8})=8\ln x}$, so ${\displaystyle f'(x)=8{\frac {1}{x}}={\frac {8}{x}}}$

c) ${\displaystyle f(x)=\ln(4-x^{2})}$

Solution:
${\displaystyle f'(x)={\frac {1}{4-x^{2}}}(4-x^{2})'={\frac {1}{4-x^{2}}}(-2x)={\frac {-2x}{4-x^{2}}}}$