User:Dfeuer/Derivative of P-Norm wrt P

Theorem
For each $p \in \R_{\ge 1}$, let $\ell^p$ denote the $p$-sequence space.

Let $\mathbf x$ be a sequence of complex numbers.

Suppose that for some non-degenerate open interval $U$, $p \in U \implies \mathbf x \in \ell_p$.

Let $f:U \to \R$ be defined by:
 * $f(p) = \left\Vert{ \mathbf x }\right\Vert_p$

where $\left\Vert{ \mathbf x }\right\Vert_p$ is the $p$-norm of $\mathbf x$, so
 * $f(p) = \left({\sum_{i=1}^\infty |x_i|^p}\right)^{1/p}$

Then $D_p f(p) = $

Proof
By the general definition of exponentiation,
 * :$\displaystyle f(p) = \exp \left({\frac 1 p \ln \sum_{i=1}^\infty |x_i|^p}\right)$

So $D_p f(p) = f(p) D_p \left({\frac 1 p \ln \sum_{i=1}^\infty |x_i|^p}\right)$.