5.3 The Chain Rule
With the above lemma, it is easy to prove the chain rule.
Theorem 5.3.1 (The chain rule) Let U and V be open sets U ⊆ X and V ⊆ Y .
Suppose f : U → V is differentiable at x ∈ U and suppose g : V → Fq is differentiable at
∈ V . Then g ∘ f is differentiable at x and
Proof: This follows from a computation. Let B
and let r
also be small enough
, it follows that f
. Such an r
exists because f
is continuous at x
, the definition of differentiability of g
By Lemma 5.2.3
. From the definition of the derivative D
exists and equals