Calculate the differential at a point I'm to calculate the differential to $f(x_1,x_2)=e^{x_1}+x_2$ at $x=(2,-1)$
I get the differential to be: $e^{x_1}dx_1+dx_2$
I'm ready to plug in the coordinates (only $x_1$) but I'm put off by the "$dx_1$" and "$dx_2$", how do I deal with them if I only want a value as an output?

For $f\colon \mathbb R^n\to \mathbb R$, and $x\in \mathbb R^n$, you can compute the differential of $f$ at $x$, $df_x \colon \mathbb R^n\to \mathbb R$ as follows. 
If $\nabla f$ is the gradient of $f$, then $df_x$ is given by $$d f_{x}(v)= (\nabla f)_{x}\cdot v$$
Then, for $f(x_1,x_2)=e^{x_1}+x_2$ and $x=(2,-1)$ you have that $(\nabla f)_x=\left(\frac{\partial f}{\partial x_1}(x), \frac{\partial f}{\partial x_2}(x)\right)=(e^2, 1)$. And for any $v=(v_1, v_2)\in \mathbb R^2$, you have that
$$df_x(v)=e^2v_1+v_2.$$ 
