Differentiability in R^n If $f: U \rightarrow \mathbb R^n$ ($U \subset \mathbb R^m$ is an open set) is differentiable and $f(x) \neq 0$ $\forall x \in U$ $\Rightarrow$ $\varphi: U \rightarrow \mathbb R$, $\varphi(x) = \frac {1}{||f(x)||}$ is differentiable.
I know how to show that $\varphi$ is differentiable, but I'm having problems to find the differential $\varphi'(x).v$, $\forall v \in \mathbb R^m$.

Consider first $\phi(x)=\|f(x)\|^2=\sum_i f_i(x)^2$ so that $\frac{\partial \phi}{\partial x_j}=2 \sum_i f_i(x) \frac{\partial f_i}{\partial x_j}(x)$. Now, observe $\varphi(x)=\frac{1}{\sqrt{\phi(x)}}$ and apply the chain rule:
$$\frac{\partial \varphi}{\partial x_j}=\frac{d \varphi}{d \phi}\frac{\partial \phi}{\partial x_j}=-\frac{1}{2}\frac{1}{(\phi(x)^{\frac{3}{2}}}\frac{\partial \phi}{\partial x_j}=-\frac{1}{\|f(x)\|^3} \sum_i f_i(x) \frac{\partial f_i}{\partial x_j}(x)$$
