Question

If $f\left(1\right)=4,f\text{'}\left(1\right)=2$, find the value of the derivative of $\log \left(f\left(e^x\right)\right)$ w.r. to x at the point x = 0.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have},f\left(1\right)=4\mathrm{and}f\text{'}\left(1\right)=2 \\ \mathrm{Let}y=\log \left\{f\left(e^x\right)\right\} \end{gathered}$$

$$\begin{gathered} \Rightarrow \frac{dy}{dx}=\frac{d}{dx}\left[\log \left\{f\left(e^x\right)\right\}\right] \\ \Rightarrow \frac{dy}{dx}=\frac{1}{f\left(e^x\right)}\times \frac{d}{dx}\left\{f\left(e^x\right)\right\} \\ \Rightarrow \frac{dy}{dx}=\frac{1}{f\left(e^x\right)}\times f\text{'}\left(e^x\right)\times \frac{d}{dx}\left(e^x\right) \\ \Rightarrow \frac{dy}{dx}=\frac{e^{x}f\text{'}\left(e^x\right)}{f\left(e^x\right)} \\ \mathrm{Putting}x=0,\mathrm{we}\mathrm{get}, \\ \frac{dy}{dx}=\frac{e^{0}f\text{'}\left(e^0\right)}{f\left(e^0\right)} \\ \Rightarrow \frac{dy}{dx}=\frac{1f\text{'}\left(1\right)}{f\left(1\right)} \\ \Rightarrow \frac{dy}{dx}=\frac{2}{4}\left[\because f\text{'}\left(1\right)=2\mathrm{and}f\left(1\right)=4\right] \\ \Rightarrow \frac{dy}{dx}=\frac{1}{2} \end{gathered}$$