Question

Derivative of

$\ln \left(f\right(x\left)\right)$

Answer 1

The derivative of$\ln \left(x\right)is\frac{1}{x}.$

Though both$\log x$ and $\ln x$ are logarithms, their derivatives are NOT the same. i.e.,
$$\begin{gathered} \frac{d}{dx}(\ln x)=\frac{1}{x} \\ \frac{d}{dx}(\log x)=\frac{1}{x(\ln 10)} \end{gathered}$$

We know that the domain of $\ln \left(x\right)isx>0$ and

thus, $\frac{d}{dx}(\ln |x\left|\right)=\frac{1}{x}$ as well.

Hence, the Derivative of $\ln \left(f\right(x\left)\right)$ using the chain rule is $\frac{1}{\left(f\right(x\left)\right)}\times f\text{'}\left(x\right)$