Derivative of \|x\|^p

What is the derivative of the function x\mapsto\|x\|^p where x\in\boldsymbol{{R}}^n? It was the last question my friend Peter Carr asked me before he left for the math conference in the sky.

The answer is D\|x\|^p = p\|x\|^{p - 2}x^*, but the notation needs some explaining. It looks similar to dx^p/dx = px^{p-1} but the definiton of derivative of a function from a vector space to a vector space is the Frechet derivative.

If f\colon\boldsymbol{{R}}\to\boldsymbol{{R}} then its derivative is Df(x) = f'(x) = df(x)/dx = \lim_{h\to 0} (f(x + h) - f(x))/h when it exists. If F\colon\boldsymbol{{R}}^n\to\boldsymbol{{R}}^m then we cannot use this definition f(x + h) - f(x)\in\boldsymbol{{R}}^m and h\in\boldsymbol{{R}}^n so we cannot “divide” the two vectors.