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Differentiate an integral (Lebesgue integral)

Let $f:[0,1]\to\mathbb{R}$ be a bounded (Lebesgue) measurable function. Consider the function $$w(p)=\int_0^1|f|^p\,d\mu$$. Is $w(p)$ differentiable at any $0<p<\infty$? I.e. does $w'(p)$ exist for …