$ \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\abs}[2][]{\left\lvert#2\right\rvert_{\text{#1}}} \newcommand{\ket}[1]{\left\lvert#1 \right.\rangle} \newcommand{\bra}[1]{\langle\left. #1\right\rvert} \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\dd}{\text{d}} \newcommand{\dv}[2]{\frac{\dd #1}{\dd #2}} \newcommand{\pdv}[2]{\frac{\partial}{\partial #1}} $
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Partial Derivative

Let $A\subseteq\mathbb{R}^n$, $B\subseteq\mathbb{R}^m$, $f:A\to B$ and $g:B\to\mathbb{R}^l$. Suppose $\vec{a}\in \text{int}(A)$ and $\vec{b}=f(\vec{a})\in \text{int}(B)$. If $f$ is differentiable at $\vec{a}$ and $g$ is differentiable at $f(\vec{a})$, then the composition $h:=g\circ f$ defined by $h(\vec{x})=g(f(\vec{x}))$ is differentiable at $\vec{a}$ and the derivative is $$D_{h}(\vec{a})=Dg(f(\vec{a}))\circ D_f(\vec{a})​$$

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Partial Derivative

Let $A\subseteq\mathbb{R}^n$, $B\subseteq\mathbb{R}^m$, $f:A\to B$ and $g:B\to\mathbb{R}^l$. Suppose $\vec{a}\in \text{int}(A)$ and $\vec{b}=f(\vec{a})\in \text{int}(B)$. If $f$ is differentiable at $\vec{a}$ and $g$ is differentiable at $f(\vec{a})$, then the composition $h:=g\circ f$ defined by $h(\vec{x})=g(f(\vec{x}))$ is differentiable at $\vec{a}$ and the derivative is $$D_{h}(\vec{a})=Dg(f(\vec{a}))\circ D_f(\vec{a})​$$

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definition
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FullPage
definition
concepts
used in
hypothesis
results

Partial Derivative

Let $A\subseteq\mathbb{R}^n$, $B\subseteq\mathbb{R}^m$, $f:A\to B$ and $g:B\to\mathbb{R}^l$. Suppose $\vec{a}\in \text{int}(A)$ and $\vec{b}=f(\vec{a})\in \text{int}(B)$. If $f$ is differentiable at $\vec{a}$ and $g$ is differentiable at $f(\vec{a})$, then the composition $h:=g\circ f$ defined by $h(\vec{x})=g(f(\vec{x}))$ is differentiable at $\vec{a}$ and the derivative is $$D_{h}(\vec{a})=Dg(f(\vec{a}))\circ D_f(\vec{a})​$$

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Partial Derivative

Let $A\subseteq\mathbb{R}^n$, $B\subseteq\mathbb{R}^m$, $f:A\to B$ and $g:B\to\mathbb{R}^l$. Suppose $\vec{a}\in \text{int}(A)$ and $\vec{b}=f(\vec{a})\in \text{int}(B)$. If $f$ is differentiable at $\vec{a}$ and $g$ is differentiable at $f(\vec{a})$, then the composition $h:=g\circ f$ defined by $h(\vec{x})=g(f(\vec{x}))$ is differentiable at $\vec{a}$ and the derivative is $$D_{h}(\vec{a})=Dg(f(\vec{a}))\circ D_f(\vec{a})​$$

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