Let $f\left(A\right)=e^{A^{2}}$ where $A$ is an $n\times n$ matrix. Show that $f$ is differentiable and compute its derivative.
I know this is kind of a basic question, but I am not sure how to solve it. By definition, we want to show that there exists a matrix $B$ such that $\lim_{H\to0}\frac{\left|f\left(A+H\right)-f\left(A\right)-BH\right|}{\left|H\right|}=0.$ Is this right? But since we're working with matrices I am not sure what to do. Any guidence would be appreciated.