On the chain rule

Question

I am following this tutorial, and I couldn't understand 2 mathematical rules, I am looking for a very simple, step-by-step logic please:

The Power Rule: $\frac{d}{dx} u^n = nu^{n-1}\frac{du}{dx}$

The Chain Rule: $\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$

P.S And yes, I did search but could not find fair explanation.

Answer 1

If $u=u(x)$ then \begin{align} \frac{d}{dx}u^n &= \frac{d}{du}u^n\frac{du}{dx}\\ &= n u^{n-1}\frac{du}{dx} \end{align}

Also, let $y=g(x)$ then \begin{align} \frac{d}{dx}f(y) &= \frac{dy}{dx}\frac{d}{dy}f(y) \\ &= g'(x)f'(y)\\ &=g'(x)f'(g(x)) \end{align}

Reference

Answer 2

I'll use Taylor expansion to define the derivative: $f'$ is the only number (/linear application) such that: $f(x+h)-f(x)-f'(x)h=\omicron(h)$

One have: $f(g(x+h))=f(g(x)+g'(x)h+\omicron(h))=f(g(x))+f'(g(x))\bigl(g'(x)h+\omicron(h)\bigr) +\omicron(h)$

Hence the desired result: $f(g(x+h))-f(g(x))-f'(g(x))g'(x)h=\omicron(h)$