Help! Vector-valued functions.

define $f:\mathbb{R}^2\rightarrow\mathbb{R}^3$ by $f(u,v)=(u^2-5v,ve^{2u},2u-\log(1+v^2))$. Suppose $g:\mathbb{R}^2\rightarrow \mathbb{R}^2$ is of class $C^1$, $g(1,2)=(0,0)$ and $$Dg(1,2)=\left[\begin{matrix} 1 & 2 \ 3 & 5\end{matrix}\right]$$Compute $D(f\circ g)(1,2)$

Question

Help! Vector-valued functions.

define $f:\mathbb{R}^2\rightarrow\mathbb{R}^3$ by $f(u,v)=(u^2-5v,ve^{2u},2u-\log(1+v^2))$. Suppose $g:\mathbb{R}^2\rightarrow \mathbb{R}^2$ is of class $C^1$, $g(1,2)=(0,0)$ and $$Dg(1,2)=\left[\begin{matrix} 1 & 2 \ 3 & 5\end{matrix}\right]$$Compute $D(f\circ g)(1,2)$

richyw · Answer

so I know that$$Df=\left[\begin{matrix} 2u & -5 \ 2ve^{2u} & e^{2u} \ 2 & -\frac{2v}{1+v^2}\end{matrix}\right]$$

richyw · Answer

why wouldn't this just be $$Df(1,2)\cdot Dg(1,2)$$

kinggeorge · Answer

If you were looking for $D(f+g)(1,2)$ you could simplify as $Df(1,2)+Dg(1,2)$, but differentiation is only linear with respect to addition. If you multiplied your functions together, you need to use the chain rule.