Prove: $\nabla_w((\textbf{w}^T\textbf{x})^2) = 2\textbf{w}^T\textbf{x}\textbf{x}^T$

Question

Consider the following two $\mathbb{R}^n$ vectors w = $[w_0, w_1, ... , w_{n-1}]$, x = $[x_0, x_1, ..., x_{n-1}]^T$.

I proved that $\nabla_w(\textbf{w}^T\textbf{x}) = \textbf{x}$.

I also proved that $\nabla_w((\textbf{w}^T\textbf{x})^2) = 2\textbf{x}\textbf{w}^T\textbf{x}$.

Now I need to prove that $\nabla_w((\textbf{w}^T\textbf{x})^2) = 2\textbf{w}^T\textbf{x}\textbf{x}^T$.

I've tried solving this for quite a while to no avail. Would really appreciate some help here.