<s>[INST] How to differentiate $\left(x^2 + \frac{1}{x^2}\right)^5$ How can I differentiate this expression
$$\left(x^2 + \frac{1}{x^2}\right)^5$$
I will appreciate any help.
 [/INST] Start with the Chain Rule:$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$
where $f(x)=x^5$ and $g(x)=x^2+\frac{1}{x^2}$. $$\\$$
For  $f'(g(x))$ we get:
$$f'(x)= 5(x^2+\frac{1}{x^2})^4.$$
For $g'(x)$ we can separate the expression:
$$\frac{d}{dx} (x^2) + \frac{d}{dx} (\frac{1}{x^2})$$
$\frac{d}{dx} (x^2) = 2x$ and $\frac{d}{dx} (\frac{1}{x^2}) = \frac{-2}{x^3}$.
So, For $g'(x)$ we get:
$$2x-\frac{2}{x^3}.$$
Thus, we get  $f'(g(x))g'(x)$ as:
$$(5(x^2+\frac{1}{x^2})^4)(2x-\frac{2}{x^3}).$$
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