Derivatives of logarithmic functions

Statement

Compute the derivatives:

• (a) $f(x)=\ln(3x^2+5)$
• (b) $g(x)=x^2\ln x$, then compute $g'(e)$
• (c) $h(x)=\ln(\sin 2x)$

Solution

(a) Chain rule: $f'(x)=\dfrac{6x}{3x^2+5}$.

(b) Product rule: $g'(x)=2x\ln x+x^2\cdot\dfrac1x=2x\ln x+x$. Hence $g'(e)=2e\cdot 1+e=3e\approx 8{.}155$.

(c) $h'(x)=\dfrac{2\cos 2x}{\sin 2x}=2\cot 2x$.

$\boxed{f'=\tfrac{6x}{3x^2+5};\ g'=2x\ln x+x,\ g'(e)=3e;\ h'=2\cot 2x}$