Proving the derivative of a function of another function at a point
To do a formal proof, we can just use the chain rule.
\begin{align} \left[ \frac{d}{dx} f(ax + b) \right]_{x = x_0} &= \left[f'(x) \frac d{dx}(ax+b) \right]_{x=ax_0+b} \\ &= \lim_{x \to x_0} \frac{f(ax + b) - f(ax_0 + b)}{a(x-x_0)} a \\ &= \lim_{x \to x_0} \frac{f(ax + b) - f(ax_0 + b)}{(x-x_0)} \end{align}
The first line is the chain derivative, the second line is the definition of the derivative.