I have this function:
$f(x,y)=\left ( sin(x)y,e^{xy} \right)$
I need to prove this function is differentiable (and find the derivative) using the formal definition. However, I wasn't able to prove this for the "second function": $e^{xy}$. I started with writing:
$f(\vec{x}+\vec{h})-f(\vec{x})=e^{(x+h_{1})(y+h_{2})}-e^{xy}$.
I know the Jacobian is: $\begin{pmatrix} ye^{xy} & xe^{xy} \end{pmatrix}$, so I should reach something like: $e^{(x+h_{1})(y+h_{2})}-e^{xy}=ye^{xy}h_{1}+xe^{xy}h_{2}+o(\vec{h})$, but I have know idea how to do so. Any suggestions?
To expand on my comment: \begin{align} e^{(x+h_1)(y+h_2)}-e^{xy}&=e^{xy}(e^{h_1y+h_2x+h_1h_2}-1) \\\\ &=e^{xy}(1+h_1y+h_2x+O(h_1h_2)-1)\\\\&=ye^{xy}h_1+xe^{xy}h_2+O(h_1h_2)\end{align}