Check if function is partially differentiable by simply deriving it partially

Question

I have to check for a given point $\vec{a}$, whether a given function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is (partially) differentiable in this point. My first idea was to use the definition of the partial derivative and check if $$\lim_{h \rightarrow 0} \frac{f\left(\vec{a} + h \vec{e_k}\right)-f\left(\vec{a}\right)}{h} $$ exists. My second idea was, since this quotient can be very difficult to "break", to simply derive the function given the known rules for differentiation and check if $\vec{a}$ is part of the domain of $\partial_k f$. My questions is whether this solution is also correct.