TITLE: Finding the derivative of the norm
QUESTION [5 upvotes]: Consider function from Hilbert space to real numbers. $F(x)=\| Ax\|$. My question how to find it's derivative $F'(x)$.

REPLY [5 votes]: I assume that $A$ is a bounded linear operator on a Hilbert space $X$, and we want to compute Frechet derivative. Consider two functions
$$
G:X\to\mathbb{R}: x\mapsto\Vert x\Vert=\sqrt{\langle x,x\rangle}
$$
$$
H:X\to X:x\mapsto A(x)
$$
One may show that $G'(x)(h)=\frac{\langle x, h\rangle}{\Vert x \Vert}$ (for $x=0$, this derivative doesn't exist) and $H'(x)(h)=A(h)$ where $h\in X$. Then 
$$
F'(x)(h)=(G(H(x)))'(h)=(G'(H(x))\circ H'(x))(h)=G'(H(x))(H'(x)(h))=
$$
$$
G'(H(x))(A(h))=\frac{\langle A(x), A(h)\rangle}{\Vert Ax\Vert}
$$