Chain Rule and Gradients

Theorem

• If $c:\mathbb{R}\to {\mathbb{R}}^k$ is a Path
• If $f:{\mathbb{R}}^k\to \mathbb{R}$ is a Function
• Then, $f\circ c:\mathbb{R}\to \mathbb{R}$
• Then, in this case $\frac{d(f\circ c)}{dt}=Df(c(t))Dc(t)=\nabla f(c(t))\cdot c^{\prime }(t)$