3.6 – The Chain Rule

The Chain Rule If f and g are both differentiable and F = f • g is the composite function defined by F(x) = f(g(x)), then F is differentiable and F’ is given by the product F’(x) = f’g(x)g’(x) In Leibniz notation, if y = f(u) and u = g(x) are both differentiable functions then

Example 1 Find F’(x) if

Example 2 Differentiate (a) y = sin(x2) (b) y = sin2x

Chain Rule + Power Rule If y = [g(x)]n, what is y’ ? The Power Rule Combined with the Chain Rule If n is any real number and u = g(x) is differentiable, then

Example 3 Differentiate

Example 4 Differentiate

Example 5 Differentiate