I'm looking to find the answer and method to the following derivatives:
Let $u,\phi$ be smooth scalar functions in $\mathbb{R}^n$, and $\epsilon\in\mathbb{R}$, $p\in\mathbb{N}$. Find $$\frac{d}{d\epsilon}\left(|\nabla(u+\epsilon\phi)|^p\right)$$ $$\nabla \cdot (|\nabla u|^{p-2}\nabla u)$$
My attempt:
Let $F(\epsilon)=\nabla(u+\epsilon\phi)$. Then $$\frac{d}{d\epsilon}\left(|\nabla(u+\epsilon\phi)|^p\right)=\frac{d}{d\epsilon}\left((F(\epsilon)\cdot F(\epsilon))^{p/2}\right)=\frac{p}{2}(F\cdot F)^{\frac{p-2}{2}}2 F'\cdot F=p|\nabla(u+\epsilon\phi)|^{p-2}(\nabla u\cdot\nabla\phi+\epsilon|\nabla\phi|^2)$$
Is this right? The other, using Einstein convention: $$\nabla \cdot (|\nabla u|^{p-2}\nabla u)=\partial_i\left(((\partial_j u)(\partial_ju)^{\frac{p-2}{2}}\right)\partial_iu=\frac{p-2}{2}\left(((\partial_ju)(\partial_ju))^{\frac{p-4}{2}}\cdot 2(\partial_ju)(\partial_j\partial_iu)\right)\partial_iu=(p-2)|\nabla u|^{p-4}(\partial_ju)(\partial_j\partial_iu)(\partial_iu)$$ But I'm stuck here