Let $c_1,c_2,\ldots,c_k$ be a collection of $k$ column vecotrs in $\mathbb{R}^n$ and let $f: \mathbb{R}^n \to \mathbb{R}$. We are interested in \begin{align*} c_1^T \nabla c_2^T \nabla c_3^T \nabla c_4^T \nabla ... c_k^T \nabla f(x) \end{align*} where $\nabla$ is the gradient operator.
Question: Can this be written more compactly? Perhaps by using Hadamard product or some other matrix form.