Identity involving the Möbius function and the first derivative of the Riemann zeta function

Question

Working on the derivatives of the Riemann zeta function, I noted that, for any positive integer $n>1$, the following identity holds:

$$\frac{\zeta'(n)}{\zeta^2(n)}=\sum_{x=1}^\infty \mu(x) \frac{\log{x}}{x^n}$$

where $\mu$ is the Möbius function. I googled to find a proof or at last a citation of this formula, but did not find anything. My first idea to prove it was to start from some established identities involving the logarithmic derivatives of the zeta function (e.g., that relating the first derivative to the Von Mangoldt function) or the prime zeta function (e.g., that expressing it in terms of the Möbius function and the zeta function). However, I suppose that a relatively simple demonstration of this could be obtained using other simple approaches (for example, a direct application of the Möbius inversion formula). I would be interested to have confirmation of this.

Answer 1

It is well known that $$\frac1{\zeta(n)}=\sum^\infty_{x=1}\frac{\mu(x)}{x^n}$$

Now differentiate both sides and also use the formula $\left(\frac1f\right)’=-\frac{f’}{f^2}$