Chain Rule and Vector valued functions?

Let's recap some things: For general $f=(f_1,\ldots,f_m):U\subseteq\mathbb{R}^n\to\mathbb{R}^m$ (where $U$, the domain of $f$, is open in $\mathbb{R}^n$) and $x\in U$, $Df(x)$ is the Jacobian (or rather the linear transformation associated to it), given by $$Df(x)=\left[\frac{\partial f_i}{\partial x_j}(x)\right]_{\substack{i=1,\ldots,m\\j=1,\ldots,n}}$$ (wherever this makes sense). From this definition, the following are obvious:

• When $m=1$, $Df(x)$ is simply a row, and in fact it is equal to $\nabla f(x)$, the gradient of $f$ at $x$.

• When $n=1$, $Df(x)$ is a column, where each entry is simply the derivative of $f_j$ at $x$.

• When $m=n=1$, $Df(x)$ is a number, equal to $f'(x)$.

Now recall the Chain Rule:

For $f:U\subseteq \mathbb{R}^n\to V\subseteq\mathbb{R}^m$ and $g:V\subseteq\mathbb{R}^m\to\mathbb{R}^p$, $$D(g\circ f)(x)=Dg(f(x))Df(x),$$ where the RHS is simply product of matrices.

Now to your problem: to use the chain rule, you have to see $f$ as a composition of two simpler functions, which are differentiable everywhere. If we put $h:\mathbb{R}^n\to(0,\infty)$ as $h(x)=\Vert x\Vert^2$, and $k:(0,\infty)\to\mathbb{R}$ by $k(t)=\frac{t^2}{1+t}$, we have $f=k\circ h$, and it is easier to calculate $Dh$ and $Dk=k'$.

I'll leave the details to you, but here's what you should get: For $x=(x_1,\ldots,x_n)$, use the definition of $Dh(x)$ to find $Dh(x)=2[x_1\ x_2\cdots x_n]$.

For $k$, one-variable calculus gives $k'(t)=\frac{t^2+2t}{(1+t)^2}$.

By the chain rule, $$Df(x_1,\ldots,x_n)=2\frac{(\Vert x\Vert^4+2\Vert x\Vert^2)}{(1+\Vert x\Vert^2)^2}[x_1\cdots x_n]$$

I assume that you are working with the euclidean norm. You are right that these are length n (1 by n or n by 1 depending on convention) vectors. So we have by definition of the norm

$f(x)=\frac{||x||^4}{1+||x||^2}=\frac{\sqrt{(x_{1}^{2}+x_{2}^{2}+....+x_{n}^{2})}^4}{1+\sqrt{(x_{1}^{2}+x_{2}^{2}+....+x_{n}^{2})}^2}= \frac{(x_{1}^{2}+x_{2}^{2}+....+x_{n}^{2})^2}{1+(x_{1}^{2}+x_{2}^{2}+....+x_{n}^{2})}$

Just reducing exponents. Can you take it from here using the chain rule?