derivative of sum of vectors

Question

suppose i need to make the partial derivative of this vector function $f(\vec{a},\vec{b})=\frac{1}{| \vec{a}+\vec{b}|}$ respect to $\vec{a}$:

$\frac{\partial }{\partial \vec{a}} f(\vec{a},\vec{b})$,

this should be equal to:

$-\frac{1}{| \vec{a}+\vec{b}|^3}(\vec{a}+\vec{b})$,

is the result correct? Or something like:

$-\frac{\hat{a}}{| \vec{a}+\vec{b}|^2}$

Answer 1

Write out the function fully. If $a=(a_1,a_2,\dots,a_n)$, then $$f(\vec a,\vec b) = \frac{1}{\sqrt{\sum_{i=1}^n(a_i + b_i)^2}}.$$

Now calculate $\frac{df}{da_i}$ for each $i$.