Using $e^{-\lambda w_1}[1+\lambda(w_2-w_1)e^{-\lambda(w_2-w_1)}$ take the second derivative to find the joint density function $\lambda^2e^{-\lambda w_2}.$
It sounds silly but I'm having issues in taking this derivative.
I realise this simplifies to $e^{-\lambda w_2}[1+\lambda(w_2-w_1)].$
When I take the second derivative I get $e^{-\lambda w_2}(w_2-w_1)+\lambda e^{-\lambda w_2}[1+\lambda(w_2-w_1)]$ first and then a result of $\lambda e^{-\lambda w_2}(w_2-w_1)+\lambda e^{-\lambda w_2}(w_2-w_1)+\lambda^2e^{-\lambda w_2}[1+\lambda(w_2-w_1)]$ but this should simplify to $\lambda^2e^{-\lambda w_2}$.
I shortly found out the way to do this is to first take the partial derivative with respect to $w_1$ and then the one with respect to $w_2$.
Again rewrite the function as $e^{-\lambda w_2}+\lambda w_2 e^{-\lambda w_2}-\lambda w_1 e^{-\lambda w_2}$. Following the derivative strategy mentioned above this becomes $-\lambda e^{-\lambda w_2}$ then $\lambda^2 e^{-\lambda w_2}$.