Is the softmax Lipschitz differentiable?

Question

For $j=1,...,J$, let $f_j : R^n \to R$ be given by

$$f_j(x) = \frac{e^{x_j}}{\sum_{i=1}^J e^{x_i}}.$$

So $f_j$ is the $j$th component of the softmax function. Is this function Lipschitz differentiable? That is, does there exist $L>0$ such that

$$||\nabla f_j(x) - \nabla f_j(y)|| \leq L ||x - y||$$

for all $x, y \in R^n$? If so, how good an estimate of $L$ can we get?

I know that $f_j$ is Lipschitz-differentiable in the case that $n=2$, because the eigenvalues of $\nabla^2 f_j$ have a closed form solution. But I'm not sure how to prove the general case.