Summary: We study the large time asymptotic behavior, in \(L^p\) \((1\leq p\leq \infty)\), of higher derivatives \(D^\gamma u(t)\) of solutions of the nonlinear equation \[ \begin{cases} u_t+{\mathcal T}u=a\cdot\nabla^\theta(\psi(u)) \quad\text{on }\mathbb{R}^n \times(0,\infty),\\ u(0)=u_0\in L^1(\mathbb{R}^n), \end{cases} \] where the integers \(n\) and \(\theta\) are bigger than or equal to 1, \(a\) is a constant vector in \(\mathbb{R}^p\) with \(p={\theta+n-1 \choose n-1}=\frac{(\theta+n-1)!}{\theta!(n-1)!}\). The function \(\psi\) is a nonlinearity such that \(\psi\in{\mathcal C}^\theta(\mathbb{R})\) and \(\psi(0)= 0\), and \({\mathcal T}\) is a higher order elliptic operator with nonsmooth bounded measurable coefficients on \(\mathbb{R}^n\). We also establish faster decay when \(u_0\in L^1(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n)\).