Simple explanation of the differentiation of $\ln(f(x))$ Could somebody explain why the derivative of $\ln[f(x)]$ = $f'(x)/f(x)$ .
Why is it not simply $1/f(x)$ as is the case for the derivative of $\ln(x)$ being $1/x$? 

The derivative of $\log(f(x))$ with respect to $x$ is given by the chain rule.  Let $y=f(x)$.  
Noting that the derivative of $\log(y)$ with respect to $y$ is $\frac1y$, we can write
$$\begin{align}
\frac{d}{dx}\log(f(x))&=\left(\left.\frac{d\log(y)}{dx}\right)\right|_{y=f(x)}\\\\
&=\left(\left.\frac{d\log(y)}{dy}\frac{dy}{dx}\right)\right|_{y=f(x)}\\\\
&=\left.\left(\frac1y \frac{dy}{dx}\right)\right|_{y=f(x)}\\\\
&=\frac{1}{f(x)}\frac{df(x)}{dx}\\\\
&=\frac{f'(x)}{f(x)}
\end{align}$$
