Let $d\in\mathbb N$, $\Omega\subseteq\mathbb R^d$ be bounded and open with smooth boundary and $(\lambda_1,e_1)$ denote the first eigenpair of the Dirichlet Laplacian in $\Omega$, i.e. $\lambda_1>0$ and $e_1\in C^2(\overline\Omega)$ with \begin{align}\left.e_1\right|_\Omega&>0;\tag1\\\left.e_1\right|_{\partial\Omega}&=0\tag2\end{align} and $$-\Delta e_1=\lambda_1e_1\;\;\;\text{in }\Omega\tag3.$$
Let $$c:=\max_{\overline\Omega}e_1$$ and $$\theta:=\frac{e_1}c.$$
Let $p>0$. I've read that $$p(p-1)\theta^{p-2}+\Delta\theta^p=p\theta^{p-1}\Delta\theta\tag4,$$ but I don't get why this formula holds. If $f\in C^2(\Omega)$, then $$\Delta f^p(x)=(p-1)pf^{p-2}(x)\|\nabla f(x)\|^2+pf^{p-1}(x)\Delta f(x)\tag5$$ for all $x\in\Omega$. So, in comparison with $(5)$, $(4)$ is missing a factor of $\|\nabla\theta\|^2$ in the first summand. Is my formula $(5)$ wrong or do the assumptions somehow imply that $\|\nabla\theta\|=1$?