<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] You have $\frac{(1+\sqrt{5-x^2})(\sqrt{5-x^2})+x^2}{x(\ln 10)(1+\sqrt{5-x^2})(\sqrt{5-x^2})}$
splitting into 2 fractions gives ;
$\frac{(1+\sqrt{5-x^2})(\sqrt{5-x^2})}{x(\ln 10)(1+\sqrt{5-x^2})(\sqrt{5-x^2})}+\frac{x^2}{x(\ln 10)(1+\sqrt{5-x^2})(\sqrt{5-x^2})}$
$=\frac1{x(\ln(10))}+\frac{x}{(\ln 10)(1+\sqrt{5-x^2})(\sqrt{5-x^2})}$
$ = \frac1{x(\ln(10))}+\frac{10^y}{\ln(10)(\sqrt{5-x^2})}$
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