# Frechet derivative of exp(X)

I want to show directly from the definition of exp(X) that it is Frechet differentiable at every $$X \in \mathbb{R^{n\times n}}$$. The definition being exp$$(X):= \sum \frac{X^k}{k!}$$

I know continuous partial derivatives implies the function is frechet differentiable. However, this was proven in my notes in the the context of functions from an n-dimensional Euclidean space to an m-dimensional one and so I'm not confident in extending it to matrix functions (matrix-valued). Can I take partials as: $$\frac{\partial(exp(X))}{X_{ij}}$$ where the denominator is the i-jth entry of the matrix $$X$$. Then show that these are continuous and therefore exp$$(X)$$ is differentiable at $$X$$? If yes, then this 'object' is a matrix is it not? And so is continuous iff each of it's entries are continuous? And am I correct in thinking $$\frac{\partial(X)}{X_{ij}}$$ is the matrix with a $$1$$ in the i-jth position and $$0$$ elsewhere? I could then use this to show (by product rule?) the higher order terms in the sum are partially differentiable w.r.t $$X_{ij}$$ and hence so is their sum.

Is the Jacobian matrix of derivative of exp$$(X)$$ an $$n^2 \times n^2$$ matrix? Any comments are welcome, just want someone to point out mistakes or perhaps add something you might suspect I don't quite understand. Thank you.