The Chain Rule

In words: differentiate the outside function, leave the inside alone, and multiply by the derivative of the inside. Rates multiply: if $u$ changes $3$ times as fast as $x$ and $y$ changes $2$ times as fast as $u$, then $y$ changes $6$ times as fast as $x$.

The proof starts by splitting the difference quotient. When $g(x + h) \ne g(x)$,

As $h \to 0$, $g(x + h) \to g(x)$, because $g$ is differentiable, and so continuous, at $x$. So the first factor approaches $f'(g(x))$ and the second approaches $g'(x)$. The split doesn't work when $g(x + h) = g(x)$ for values of $h$ arbitrarily close to $0$. A slightly longer argument covers that case, and the rule still holds.

The picture is the chain rule with inside function $2x$. The derivative of $\sin 2x$ is $\cos(2x) \cdot 2$.

The first and last follow from the chain rule. Since $a^x = e^{x\ln a}$, its derivative is $e^{x\ln a} \cdot \ln a = a^x \ln a$. The middle one is a constant multiple: since $\log_a x = \frac{\ln x}{\ln a}$ and $\ln a$ is a constant, its derivative is $\frac{1}{x \ln a}$. For $x < 0$, $\ln|x| = \ln(-x)$, whose derivative is $\frac{-1}{-x} = \frac{1}{x}$.