## richyw Group Title 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 $$D(f\circ g)(1,2)$$ one year ago one year ago

1. richyw

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

2. richyw

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

3. KingGeorge

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.