Derivative of a function in the context of Sobolev spaces

Question

Consider $B_1$ the unitary ball in $R^n$ centered in the origin and $2 \leq p < \infty$. Let $ \psi \in W^{1,p}(B_1)$. Let $h \in W^{1,p}(B_1) $ with $h - \psi \in W^{1,p}_{0}(B_1)$ with $\Delta_p h = 0$. Define $\psi_t = t \psi +(1- t) h , 0 \leq t \leq 1$. Consider the function

$$ f(t) = \int_{B_1} |\nabla \psi_t|^p \ dx , 0 \leq t \leq 1$$

I am reading a paper and the authors use the differentiability of $f$ . I have no idea to how to prove this. Please someone could point me a reference or give a proof?

thanks in advance

Answer 1

Isn't it the case that $\partial_t \psi_t$ exists, and thus so does

$\displaystyle \int_{B_1} \partial_t |\nabla \psi_t|^p dx \ \ = \ \ \frac{d }{dt} \int_{B_1} |\nabla \psi_t|^p dx $

Or am I missing something?