Proving a quaternion Fourier transform identity (not too complicated)

Question

So, I am getting a bit into Quaternion Fourier Transform (QFT), and I am going over an article of Eckhard M. S. Hitzer, which first definition for a QFT is the following:

\begin{equation*} \hat{f}(\textbf{u}) = \int_{\Bbb R^2} e^{-ixu}f(\textbf{x})e^{-jyv}d^2\textbf{x} \end{equation*} For $f \in L^2(\Bbb R^2,\Bbb H)$, $\textbf{x} = (x,y)$, $\textbf{u} = (u,v)$ and $\hat{f}(u) : \Bbb R^2 \rightarrow \Bbb H$.

After this, I have been presented with the following (simple) properties:

1. Left Linearity: Quat. function: $\alpha f(x) + \beta g(x) \Rightarrow $Quat. transform $ \alpha\hat{f}(u) + \beta\hat{g}(u)$.
2. x-Shift: Quat. function: $f(x-x_0) \Rightarrow $Quat. transform $ e^{-ix_0u}\hat{f}(u)e^{-jy_0v}$.
3. Part. deriv.: Quat. function: $\frac{\partial^{m+n}}{\partial x^m \partial y^n}f(x)$ $\Rightarrow$ Quat. transform: $(iu)^m\hat{f}(u)(jv)^n$.

I was able to prove properties $1$ and $2$, simply by applying simple transformations and a single variable change. How would one proceed to prove identity $3$ though?

Thanks for any help in advance.