Derivative of P-Norm wrt P

Theorem
Let $p \ge 1$ be a real number.

Let $\ell^p$ denote the $p$-sequence space.

Let $\mathbf x = \sequence {x_n} \in \ell^p$.

Let $\norm {\mathbf x}_p$ be a p-norm.

Suppose, $\norm {\mathbf x}_p \ne 0$.

Then:


 * $\ds \dfrac \d {\d p} \norm {\mathbf x}_p = \frac {\norm {\mathbf x}_p} p \paren { \frac {\sum_{n \mathop = 0}^\infty \size {x_n}^p \map \ln {\size {x_n} } } {\norm {\bf x}_p^p} - \map \ln {\norm {\bf x}_p} }$

Proof
We begin with the natural logarithm of $\norm {\mathbf x}_p$:

Multiplication by $\norm {\mathbf x}_p$ completes the proof.