If $J$ is a scalar function of two vectors $\boldsymbol{q}$ and $\boldsymbol{g}$:
$$J(\boldsymbol{q},\boldsymbol{g})=\frac{\boldsymbol{g\cdot g}}{\boldsymbol{q\cdot q}}.$$
How do I calculate the derivative $\frac{\partial J}{\partial \boldsymbol{q}}$?
What I have tried:
$$\frac{\partial J}{\partial \boldsymbol{q}}=\frac{\partial}{\partial \boldsymbol{q}}\left(\frac{\boldsymbol{g\cdot g}}{\| \boldsymbol{q} \|^2} \right)=-2\|\boldsymbol{q}\|^{-3} \boldsymbol{g\cdot g}=-2 \|\boldsymbol{q}\|\:( \boldsymbol{g\cdot g}) \|\boldsymbol{q}\|^{-4}=-2\|\boldsymbol{q}\|\: (\boldsymbol{g\cdot g}) (\|\boldsymbol{q}\|^{2})^{-2}\\=-2\|\boldsymbol{q}\|\:( \boldsymbol{g\cdot g}) (\boldsymbol{q\cdot q})^{-2}.$$
I am sure that my answer is wrong because it should have been a vector instead of a scalar.
For your reference, the answer given is
$$\frac{\partial J}{\partial \boldsymbol{q}}=-2\boldsymbol{q}( \boldsymbol{g\cdot g})/(\boldsymbol{q\cdot q})^2.$$
Thank you in advance.
The partial derivative of a scalar w.r.t. a vector is a vector. Take partial derivative w.r.t. every element of the vector and put these together to get your result. Assuming that elements of $q$ are not depending on each other or a common parameter: $$\frac{\partial J}{\partial q_i}=\frac{\partial}{\partial q_i}\left(\frac{\boldsymbol{g\cdot g}}{\| \boldsymbol{q} \|_2^2} \right)=\frac{\partial}{\partial q_i}\left(\frac{\boldsymbol{g\cdot g}}{\sum_k q_k^2} \right)$$ If $q$ and $g$ do not depend on a common parameter then $\boldsymbol{g\cdot g}$ is nothing but a constant $$\frac{\partial J}{\partial q_i}=\boldsymbol{g\cdot g}\frac{\partial}{\partial q_i}\left(\frac{1}{\sum_k q_k^2 } \right)=-\boldsymbol{g\cdot g}\left({\sum_k q_k^2 }\right)^{-2}2q_i$$
$$\frac{\partial J}{\partial \boldsymbol{q}}=-2\boldsymbol{q}\frac{\boldsymbol{g\cdot g}}{\left(\boldsymbol{q\cdot q}\right)^2}$$ Hope this helps