Consider the function $$ f(x,y) = \begin{cases}\frac{xy(x^2-y^2)}{x^2+y^2}, & (x,y)\neq(0,0)\\ 0, & \text{otherwise.}\end{cases} $$
Compute $$\frac{d^2f}{dxdy}(0,0)$$ and $$\frac{d^2f}{dydx}(0,0).$$
Why does it not have the partial symbol? When it is a 'd', doesn't that mean it is one variable only?
It should probably be $\frac{\partial^2f}{\partial x\partial y}(0,0)$ and $\frac{\partial^2f}{\partial y\partial x}(0,0)$. You should check your book (or other material or instructor) for notation conventions as they vary from source to source. In some contexts derivatives with $d$, $\partial$ and $D$ mean the same thing, sometimes not.
For example, you could be asked to show that under suitable assumptions on a function $f:\mathbb R^n\to\mathbb R$ you have $\nabla f=Df$. I can think of two meanings for such an exercise: the gradient is the same thing as the derivative as a linear approximation (if $Df$ happens to be defined so); or that the strong and weak gradients are the same thing.
Even though people use sloppy notation and you just have to live with it, you should always be sure that you know what the notation means exactly.