What is the chain rule?

A special rule, the chain rule, exists for differentiating a function of another function (finding dy/dx).

Consider the expression cos x² . We call such an expression a ‘function of a function’.  We could identify them more mathematically by saying that f(x) = cos x and g(x) = x², such that f(g(x)) = f(x²) = cos x².

How to use the chain rule:

1. substitute u = g(x), which gives y = f(u)

2. Use the chain rule: dy/dx= dy/du × du7dx. 

Example: differentiate y = cos x²

Let u = x²  and  y = cos u

du/dx = 2x and dy/du = -sin u.

Use the chain rule: dy/dx = dy/du × du/dx = -sin u × 2x = -2x sin x²