Need to calculate the gradient of the following function:
$
f(x,y) = x^2 + 2y^2 + 2\sin (2\pi x)\sin (2\pi y)
$

Here's what I got:

$
\nabla f(x,y) = 2x + 2y + 4\pi [\sin (2\pi x)\sin (2\pi y) \cos (2\pi x) \cos (2\pi y)]
$

But being as this is my first time doing gradient calculations (need it for a course and I never took calculus 3, so this is self taught) I want to make sure it's right before I move on. Can someone double check for me?

2. $\displaystyle \nabla f(x,y)=(2x+4\pi cos(2\pi x)sin(2\pi y))i+(2+4\pi sin(2\pi x)cos(2\pi y))j$