Math 22 Derivatives of Logarithmic Functions

Derivative of the Natural Logarithmic Function

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

Exercises Find the derivative of the function

a) $f(x)=\ln(7x)$

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

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

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

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

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

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