Linear Transformations · Challenge problem

Let $V$ be the $6$-dimensional space of functions spanned by $e^x\cos x$, $e^x\sin x$, $xe^x\cos x$, $xe^x\sin x$, $x^2e^x\cos x$ and $x^2e^x\sin x$. Differentiation, $D(f) = f'$, is a linear map from $V$ to $V$. Find $\det D$.