# Computing differentials of a quotient (product) of differential functions

Problem Statement: Let $f:U\rightarrow \mathbb{R}^{n}$ and $g:U\rightarrow \mathbb{R}^{p}$ be differentiable ($U\subset \mathbb{R}^{m}$ open). Define: $$h(\mathbf{x}):U\setminus \ker{f}\rightarrow \mathbb{R}^{p},$$ $$\ \ \ \ \mathbf{x}\mapsto \frac{g(\mathbf{x})}{\lVert f(\mathbf{x})\rVert},$$ with $\lVert\cdot\rVert$ the euclidean norm in $\mathbb{R}^{p}$. Compute $dh(\mathbf{x})[\mathbf{v}]$, for each $\mathbf{v}\in \mathbb{R}^{m}$

So I am trying to work this problem out, but am a little confused because I would think that I need to use the product/quotient rule to determine $dh$, but we have not learned that in my Analysis course yet. I first took $\phi(\mathbf{x})=\lVert f(\mathbf{x})\rVert=\sqrt{\langle f(\mathbf{x}),f(\mathbf{x})\rangle}$, where $\langle\ ,\ \rangle$ is the standard inner product in $\mathbb{R}^{m}$, then I computed $d\phi(\mathbf{x})[\mathbf{v}]=\frac{1}{\lVert f(\mathbf{x})\rVert}\langle df(\mathbf{x})[\mathbf{v}],f(\mathbf{x})\rangle$. I am not sure where to go from here. Should I simply apply regular the product rule with $g(\mathbf{x})$ and $\frac{1}{\lVert f(\mathbf{x})\rVert}$?

Also, when computing differentials in real analysis, must we be precise in doing so by using the $\epsilon$-$\delta$ definition of derivative? In computing $d\phi(\mathbf{x})[\mathbf{v}]$, I just applied the standard limit definition to find the directional derivative in the direction of $\mathbf{v}\in\mathbb{R}^{m}$.

Any hints to lead me in the right direction are appreciated!