Category:Derivative of Composite Function
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This category contains pages concerning 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:
- $\map {D_x} {\map f {\map g x} } = \map {D_{\map g x} } {\map f {\map g x} } \map {D_x} {\map g x}$
Subcategories
This category has only the following subcategory.
Pages in category "Derivative of Composite Function"
The following 15 pages are in this category, out of 15 total.
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- Derivative of Composite Function
- Derivative of Composite Function/3 Functions
- Derivative of Composite Function/Also known as
- Derivative of Composite Function/Also presented as
- Derivative of Composite Function/Corollary
- Derivative of Composite Function/Informal Proof
- Derivative of Composite Function/Jacobians
- Derivative of Composite Function/Proof 1
- Derivative of Composite Function/Proof 2
- Derivative of Composite Function/Second Derivative
- Derivative of Composite Function/Third Derivative