Derivative of a power of the Lp norm ($\frac{d}{dt}\Vert u+tv{\Vert }_p^p{|}_{t=0}=p\mathrm{Re}\int |u{|}^{p-2}\overline{u}vd\mu$)

For $1<p<\infty$ and $u,v$ in an Lp space, the derivative with respect to the real scalar $t$ has the displayed form; the integrand is zero where $u=0$. The pointwise chain rule gives the derivative, while the Holder inequality makes $(|u|+|v|{)}^{p-1}|v|$ integrable and permits the dominated convergence theorem. This variational derivative characterizes nearest points to closed convex sets and constructs the Lp duality density of a bounded linear functional.