How the chain rule applies to the slope of two functions.

The chain rule can be obtained by multiplying the derivative of the first function f(x) evaluated at x_1 times the derivative of the second function g(x) evaluated at f(x_1).
Try different functions for f(x). If f(x) is differentiable then g(f(x)) is also differentiable and therefore h(g(f(x))) will also be differentiable. Hence the meaning of the "chain" in the chain rule.