Derivative of $\operatorname{Log}(\operatorname{Log}(z^2))$ Please help me with this question: (i don't know how to start)

Suppose that $f(z)$ = $\operatorname{Log}(\operatorname{Log}(z^2))$. Find $f'(z)$ where it exists, and determine the set of points at which $f$ is analytic.
Well, $log(log(z^2))$ is a composite function. More in general the derivative of a composition of 3 functions $f(g(h(x)))$ is $f'(g(h(x)))\cdot g'(h(x))\cdot h'(x)$. So, in this case
$$D[log(log(z^2))] =$$
$$\frac{1}{log(z^2)}\cdot \frac{1}{z^2}\cdot 2z =$$
$$\frac{1}{log(z^2)}\cdot \frac{2}{z} $$

As regards the $differentiability$ of the function, you need to study the continuity of $f'(x)$:
the only critical points are when
\begin{cases}
log(z^2) = 0 \to  z^2 = 1 \to z=\pm 1\\
z = 0
\end{cases} so the function is differentiable if $ z \neq \pm 1$ and $z \neq 0$
Well, $log(log(z^2))$ is a composite function. More in general the derivative of a composition of 3 functions $f(g(h(x)))$ is $f'(g(h(x)))\cdot g'(h(x))\cdot h'(x)$. So, in this case
$$D[log(log(z^2))] =$$
$$\frac{1}{log(z^2)}\cdot \frac{1}{z^2}\cdot 2z =$$
$$\frac{1}{log(z^2)}\cdot \frac{2}{z} $$

As regards the $differentiability$ of the function, you need to study the continuity of $f'(x)$:
the only critical points are when
\begin{cases}
log(z^2) = 0 \to  z^2 = 1 \to z=\pm 1\\
z = 0
\end{cases} so the function is differentiable if $ z \neq \pm 1$ and $z \neq 0$