Question

Let $y=\log \left(\log \right(x\left)\right)$ .Then find the value of $e^y\frac{dy}{dx}$.

Answer 1

Compute the value of $e^y\frac{dy}{dx}$:

A function $y=\log \left(\log \right(x\left)\right)$ is given.

Raise both sides to the power of the exponent.

So,

$$\begin{gathered} e^y=e^{\log \left(\log \right(x\left)\right)} \\ \Rightarrow e^y=\log \left(x\right)\left[\because e^{logx}=x\right] \end{gathered}$$

Now, differentiate both sides with respect to $x$.

$e^y\frac{dy}{dx}=\frac{1}{x}$.

Hence, the value of $e^y\frac{dy}{dx}$ is $\frac{1}{x}$.