How exactly is this happening?

Question

I was studying Derivative and my book says if:

Then its derivative is:

I can't understand how the writer has changed the first derivative fraction into the second one. In other words, how did he simplify?

Note: I'm really really basic, so please explain in details.

Thank you.

Answer 1

Observe that $$\sqrt{(1+x^2)^3}=(1+x^2)^{3/2}$$ Thus $$\begin{align*}\left((1+x^2)^{3/2}\right)'&=\frac{3}{2}(1+x^2)^{3/2-1}(1+x^2)'\\&=\frac{3}{2}(1+x^2)^{1/2}2x=\frac{3}{\not 2}(1+x^2)^{1/2}\not2x\\&=3x\sqrt{1+x^2}\end{align*}$$

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The first equation is due to the chain rule. i.e. $$\left(f\left(g(x)\right)\right)'=f'(g(x))g'(x)$$ with $f(x)=x^{3/2}$ and $g(x)=(1+x^2)$.