Question

If $y={\log }_{a}x,\mathrm{find}\frac{dy}{dx}.$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have},y={\log }_{a}x \\ \Rightarrow y=\frac{\log x}{\log a}\left[\because {\log }_{a}b=\frac{\log b}{\log a}\right] \end{gathered}$$

$$\begin{gathered} \Rightarrow \frac{dy}{dx}=\frac{1}{\log a}\frac{d}{dx}\left(\log x\right) \\ \Rightarrow \frac{dy}{dx}=\frac{1}{\log a}\left(\frac{1}{x}\right) \\ \Rightarrow \frac{dy}{dx}=\frac{1}{x\log a} \end{gathered}$$