Question

Use a direct application of the fact that$\frac{d}{dx}\left(f^{-1}\left(x\right)\right)=\frac{1}{f\text{'}\left(f^{-1}\left(x\right)\right)}\mathrm{to}\mathrm{prove}\mathrm{that}\frac{d}{dx}\left(\ln x\right)=\frac{1}{x}$

Short answer

The given differentiation has been proved.

Step-by-step solution

Step 1. Given Information 

The given function is$f\left(x\right)=\ln x$

Step 2. Proof 

Since $y=\ln x\mathrm{and}y=e^x$are inverses, $e^{\ln x}=x$for all x in the domain $\left(0,\infty \right)\mathrm{of}y=\ln x$.

By using property of inverse, we get,

$$\begin{gathered} e^{\ln x}=x \\ \frac{d}{dx}\left(e^{\ln x}\right)=\frac{d}{dx}\left(x\right) \\ {e}^{\ln x}\frac{d}{dx}\left(\ln x\right)=1 \\ \frac{d}{dx}\left(\ln x\right)=\frac{1}{e^{\ln x}} \\ \frac{d}{dx}\left(\ln x\right)=\frac{1}{x} \end{gathered}$$

Hence, Proved.