<s>[INST] Stuck calculating the derivative of $f(x)=\log_{10}{\frac{x}{1+\sqrt{5-x^2}}}$. I have to calculate the derivative of this:
$$f(x)=\log_{10}{\frac{x}{1+\sqrt{5-x^2}}}$$
But I'm stuck. This is the point where I have arrived:
$$f'(x) = \frac{(1+\sqrt{5-x^2})(\sqrt{5-x^2})+x^2}{x(\ln 10)(1+\sqrt{5-x^2})(\sqrt{5-x^2})}$$
How can I simplify? I didn't include all the passages.
 [/INST] Since the denominator is positive (when it's defined, that is, for $-\sqrt5\le x\le \sqrt5 $), the function is only defined for $0<x\le \sqrt5 $, so you can write it as
$$
f(x)=\log_{10}x-\log_{10}(1+\sqrt{5-x^2})
$$
which should simplify the computation very much. Recall that the derivative of $\log_{10}{x}$ is
$$
\frac{1}{x\log10}
$$
(natural logarithm, write it ln if you prefer) and apply the chain rule.
You will get
$$
\frac{1}{\log10}\left(
\frac{1}{x}-\frac{1}{1+\sqrt{5-x^2}}\frac{-x}{\sqrt{5-x^2}}
\right)
$$
Then it's just simplifications, if you really want to do more than that:
$$
\frac{1}{\log10}\frac{\sqrt{5-x^2}+5-x^2+x^2}{x\sqrt{5-x^2}(1+\sqrt{5-x^2})}
=
\frac{1}{\log10}\frac{\sqrt{5-x^2}+5}{x\sqrt{5-x^2}(1+\sqrt{5-x^2})}
$$
The derivative doesn't exist at $x= \sqrt5 $.
</s>