# Finding the derivative of the norm

Consider function from Hilbert space to real numbers. $F(x)=\| Ax\|$. My question how to find it's derivative $F'(x)$.

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 TeX: It's better to use \| instead of \parallel because of spacing. Compare: $\|Ax\|$ and $\parallel Ax\parallel$. Also, you cannot write \parallelAx\parallel, since TeX then interprets \parrallelAx as a single name of a macro. You can try \parallel{}Ax\parallel or \parallel Ax\parallel. – Martin Sleziak Jan 28 '12 at 7:48

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 $$F'(x)(h)=(G(H(x)))'(h)=(G'(H(x))\circ H'(x))(h)=G'(H(x))(H'(x)(h))=$$ $$G'(H(x))(A(h))=\frac{\langle A(x), A(h)\rangle}{\Vert Ax\Vert}$$
It's still wrong: You should have $\| Ax\|$ in the denominator. – Christian Blatter Jan 28 '12 at 20:40