I am running into issues trying to find the derivation of f(z) below. Please help
$\displaystyle f(z) = e^{x^{2}+ y^{2}}[cos(2xy) + i sin(2xy)]$
Technically the word is "differentiation", not "derivation".
I presume you know that if f(z)= f(x+ iy)= u(x, y)+ iv(x,y) then
$\displaystyle \frac{df}{dz}= \frac{\partial u}{\partial x}+ i \frac{\partial v}{\partial y}= \frac{\partial v}{\partial y}- i\frac{\partial v}{\partial y}$
(The fact that those two expressions on the right are equal is the "Cauchy- Rieman" equations. You can use either one.)
Here, of course, you have
$\displaystyle u(x,y)= e^{x^2+ y^2}cos(2xy)$
$\displaystyle v(x,y)= e^{x^2+ y^2}sin(2xy)$