In my reaseach, recently,  I came across of this problem where I have to compute, analytically,  the derivative of the dominant eigenvalue of the following matrix.  

Let $D$ be a diagonal real $n \times n $ matrix $\text{diag} \{d_{1}, \dots , d_{n}\}$. Let A be an essentially nonnegative matrix (that is $a_{ij}\geq 0$, for all $i\neq j$). I is known that $r(A+D)$ is a convex function of $D$ (see ref), where $r(A+D)$ is the dominant eigenvalue of an $A+D$. 

I need to find an expression for $\frac{\partial \rho(DA)}{\partial D_{jj}}|_{D^{*}} $, where $D^{*}$ is a diagonal matrix. It would be great if someone can direct me on how to compute this derivative. 

ref: @article{cohen1981convexity,
  title={Convexity of the dominant eigenvalue of an essentially nonnegative matrix},
  author={Cohen, Joel E},
  journal={Proceedings of the American Mathematical Society},
  volume={81},
  number={4},
  pages={657--658},
  year={1981}
}

Thank you.