Let n be a negative integer and let f(x) = x_n for x ≠ 0. Note that – n is positive and if g(x) = x – n , then f(x) = (1/g)(x). Use Example 1.8 and the quotient rule of Theorem 1.7 to show that f ′(x) = nx_{n – 1}. We know that the composition of two continuous functions is continuous. A similar result holds for the composition of differentiable functions, and it is known as the chain rule.

 

Example 1.8

To illustrate the use of Theorem 1.7, let us show that for any n ∈ N, if f (x) = xⁿ for all x ∈ R, then f ′(x) = nx_{n – 1} for all x ∈ R. Our proof is by induction. When n = 1 we have f (x) = x, so that

Theorem 1.7

Suppose that f : I → R and g : I → R are differentiable at c ∈ I. Then

(a) If k ∈ R, then the function kf is differentiable at c and