Explain the Chain Rule

The chain rule is used to differentiate composite functions, ie "a function of a function". In this case we have an outer function and an inner function. For example

Differentiate f(g(x)). Here f is the outer function and g the inner. 

The derivative of this function is found by differentiating the outer function and evaluating its derivative at the point g(x) and then multiplying by the derivative of g(x):

f(g(x))' = f'(g(x))g'(x)