Differentiation of norm

source  :  http://math.stackexchange.com/questions/291318/derivative-of-the-2-norm-of-a-multivariate-function

Suppose f:Rm→Rn. Decompose into f=(f1,…,fn). Each fi is a real-valued function, i.e., fi:Rm→R. Then

g(X)=∥f(X)∥2=∑i=1nfi(X)2−−−−−−−−−√.

Therefore,

∇g(X)=12(∑i=1nfi(X)2)−12(∑i=1n2fi(X)∇fi(X))=∑ni=1fi(X)∇fi(X)∥f(X)∥2.

This matches your answer.

If you want to write in terms of the Jacobian matrix of f instead of components fi, you can:

∇g(X)=Jf(X)Tf(X)∥f(X)∥2.