A short, intuitive explanation of the Chain Rule
Allow f(x) and g(x) to be differentiable functions over some domain D. We are looking for a formula to describe the derivative of f(g(x)).
The derivative of f at any point g(x) for fixed x is f'(g(x)). However, at each point x, g is either increasing, decreasing, or constant.
How much is g(x) increasing or decreasing at each point x? Luckily, we know g(x) at each point x is changing with rate g'(x) for each point x.
We may take the sum of each point f'(g(x)) multiplied by the rate at which g(x) is changing (g'(x)) for every x in the domain of g and we get a Riemann sum. in other words, we may take the summation of f'(g(x)) * g'(x) * dx for each point x in the domain D, and by the definition of the integral (a Riemann sum over domain D) and the First Fundamental Theorem of Calculus it follows that the integral of f'(g(x)) * g'(x) dx = f(x).
(Take u = g(x) via u-substitution to see this clearly)