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Supose $p>1$ and $f,g\in L^p(\mathbb{R})$. Let $H(s)=\int_\mathbb{R}|f(x)+s\cdot g(x)|^p\mathrm{d}x$ for $s\in \mathbb{R}$. Show that $H$ is differentiable and find its derivative.

I've tried using the definition of derivative but haven't made any progress.

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It looks like there's an exponent missing in the integrand. You probably want to consider $H(s) = \int_\mathbb{R} |f(x) + s g(x)|^{\color{red}{p}}\,dx$. If so, differentiate under the integral sign (argue why this is allowed!) and use that $s \mapsto |s|^p$ has derivative $s \mapsto p|s|^{p-1} \operatorname{sign}{(s)}$. – t.b. Jan 12 '12 at 5:01
@t.b. You're right about the exponent, it's fixed now. Thanks for the hint! – Iksander Jan 12 '12 at 6:15

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