Chain rule for differentiation

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This article is about a differentiation rule, i.e., a rule for differentiating a function expressed in terms of other functions whose derivatives are known.
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Statement for two functions

Suppose f and g are functions such that g is differentiable at a point x = x_0, and f is differentiable at g(x_0). Then the composite f \circ g is differentiable at x_0, and we have:

\frac{d}{dx}[f(g(x))]|_{x = x_0} = f'(g(x_0))g'(x_0)

In terms of general expressions:

\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)

In point-free notation, we have:

(f \circ g)' = (f' \circ g) \cdot g'

where \cdot denotes the pointwise product of functions.

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