Chain Rule for Total Derivatives

Theorem

• Let $U\subset {\mathbb{R}}^n$, $V\subset {\mathbb{R}}^k$ be Open Set
• Suppose $g:U\subset {\mathbb{R}}^n\to {\mathbb{R}}^k$
• Suppose $f:V\subset {\mathbb{R}}^k\to {\mathbb{R}}^d$
• Assume $g(U)\subset V$
• Then, $f\circ g:U\subset {\mathbb{R}}^k\to {\mathbb{R}}^d$ is defined on all of $U$
• If $g$ is Differentiable at $x_0$ and $f$ is Differentiable at $g(x_0)$, then $D(f\circ g)(x_0)=Df(y_0)Dg(x_0)$