Suppose $J(\mathbf{A})$ is defined as follows $J=\text{tr}(\log \mathbf{P})$ $\mathbf{P}=\frac{e^\mathbf{A}}{\mathbf{1} \mathbf{1}' e^\mathbf{A}}$
where division, exp and log are taken pointwise, $\mathbf{1}$ is a column vector of ones and $\mathbf{A}$ is square. What's the easiest way of showing that $\mathbf{I}-\mathbf{P}$ is the gradient of $J(\mathbf{A})$?