Category:Examples of Derivatives of Composite Functions

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This category contains examples of use of Derivative of Composite Function.

Let $I, J$ be open real intervals.

Let $g : I \to J$ and $f : J \to \R$ be real functions.

Let $h : I \to \R$ be the real function defined as:

$\forall x \in \R: \map h x = \map {f \circ g} x = \map f {\map g x}$

Then, for each $x_0 \in I$ such that:

$g$ is differentiable at $x_0$
$f$ is differentiable at $\map g {x_0}$

it holds that $h$ is differentiable at $x_0$ and:

$\map {h'} {x_0} = \map {f'} {\map g {x_0}} \map {g'} {x_0}$

where $h'$ denotes the derivative of $h$.

Using the $D_x$ notation:

$\map {D_x} {\map f {\map g x} } = \map {D_{\map g x} } {\map f {\map g x} } \map {D_x} {\map g x}$