Find the tangent space and the tangent plane to the graf of the function
$$ f(x,y) = e^{xy}$$ at the point $(0,0,1)$.
In my textbook the tangent space at the point $(x_0,y_0,f(x_0,y_0))$ consists of all the vectors on the form $$ (s,t,s \frac {\partial }{\partial x}(x_0,y_0)+t\frac {\partial }{\partial y}(x_0,y_0)), \ s,t\in\mathbb{R}$$ and the tagent plane is given by the set:
$$\text{{$(x_0,y_0,f(x_0,y_0))+(s,t,s \frac {\partial }{\partial x} (x_0,y_0) + t \frac {\partial }{\partial y} (x_0,y_0))|\ s,t\in\mathbb{R}$}}$$
So far I have that: $$\frac {\partial }{\partial x} = ye^{xy}, \ \frac {\partial }{\partial y} = xe^{xy}$$ At the point $(0,0,1)$ if we evaluate the points at the partial derivatives have that: $0+0+e$
I am unsure if this is correct and how the computation should be for the tangent space and the tangent plane.
What you did is fine. It follows from it that $\frac{\partial f}{\partial x}(0,0)=\frac{\partial f}{\partial y}(0,0)=0$. Therefore, the tangent plane is $\{(s,t,1)\,|\,s,t\in\mathbb{R}\}$.