21.6.1 The Chain Rule For Functions Of One Variable
First recall the chain rule for a function of one variable. Consider the following
picture.
Here I and J are open intervals and it is assumed that g
⊆ J. The chain rule says
that if
f′ exists and
g′ exists for
x ∈ I, then the composition,
f ∘ g also has a
derivative at
x and
(f ∘ g)′(x) = f ′(g(x))g′(x).
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Recall that f ∘ g is the name of the function defined by f ∘ g
≡ f. In the
notation of this chapter, the chain rule is written as
Df (g(x))Dg (x) = D (f ∘ g)(x ). (21.8)
| (21.8) |