Question

Differentiate $y=\log \left(x\right)$

Answer 1

Assume that$y=\log \left(x\right)$

Converting this into the exponential form, we get $e^y=x$.

Now we will take the derivative on both sides of this equation with respect to $x$. Then we get

$\frac{d}{dx}\left(e^y\right)=\frac{d}{dx}\left(x\right)$

By using the chain rule,

$$\begin{gathered} e^y\frac{dy}{dx}=1 \\ \frac{dy}{dx}=\frac{1}{e^y} \end{gathered}$$

But we have $e^y=x$

Therefore,

$\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{}\mathbf{=}\frac{\mathbf{}\mathbf{1}}{\mathbf{x}}$

Thus, the derivative of $\log \left(x\right)$ to be $\frac{\mathbf{1}}{\mathbf{x}}$