Differentiate log⁡(log⁡x⁵) w.r.t x.

The correct option is 5/xlogx⁵.

To differentiate the given function with respect to $x$, we can apply the chain rule of differentiation.
Let $y=\log \left(\log x^5\right)$.

Recall the standard derivative rule:
$\frac{d}{du}\left(\log u\right)=\frac{1}{u}$

Applying the chain rule, we differentiate the outer logarithm function first, treating the inner term $\log x^5$ as our variable:
$\frac{dy}{dx}=\frac{1}{\log x^5}·\frac{d}{dx}\left(\log x^5\right)$

Next, we differentiate the inner function $\log x^5$ using the chain rule again:
$\frac{d}{dx}\left(\log x^5\right)=\frac{1}{x^5}·\frac{d}{dx}\left(x^5\right)$

Using the power rule, the derivative of $x^5$ is $5x^4$:
$\frac{d}{dx}\left(\log x^5\right)=\frac{1}{x^5}·5x^4=\frac{5}{x}$

Substituting this back into our expression for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{1}{\log x^5}·\frac{5}{x}=\frac{5}{x\log x^5}$