Can we deduce $f^p\in W^{1,1}(U)$ if $f\in W^{1,p}(U)$

Question

Assume $U$ is some open set in $\Bbb{R}^n$.

Let $f\in W^{1,p}(U)$ then we know $f^p \in L^1(U)$ ,can we further deduce $D(f^p) \in L^{1}(U)$ that is $f^p \in W^{1,1}(U)$?

This may refer to the chian rule but $t\mapsto t^p$ does not has bounded derivative.If chain rule holds then easy to see $D(f^p) \in L^1(U)$ by Holder inequality

I have found the solution here