How to express result of derivative of a matrix? For example, if we are asked to find $f'(x)$ of $f(x) = e^{x_1 + x_2} , x \in \mathbb{R}^{2} $ then our answer would have two partial derivatives. Would we say $\frac{\partial}{\partial x_1}$ = .. and $\frac{\partial}{\partial x_2}$ = .. ? What if $x \in \mathbb{R}^{d}$ for some d?

If you have a map $f: \mathbb{R}^m \to \mathbb{R}^n$, the derivative will be the Jacobian, which is a $n$ by $m$ matrix, where the $i,j$-th coordinate contains the partial derivative of the $i$-th coordinate of $f$ with respect to the $j$-th variable. 
If $n=1$, you get the gradient (which is a vector).
An aside: Matrix calculus is a neat way of keeping track of some of the multivariable derivatives you may encounter, but its no different than just the standard calculus derivatives for $f: \mathbb{R}^m \to \mathbb{R}^n$ under the hood. 
