Hey there :)
Here's the answer with no taste.
# f_x(x, y) = 3cos(3x) # and # f_y(x, y) = -5sin(5y) #
Now the juicy parts.
To find # f_x(x, y) #, fix all other variables as constants, in this case just #y#.
So for # f_x(x, y) # we know that #cos(5y)# is only a function of #y# so we differentiate it like a constant since #y# is fixed. So we just differentiate #sin(3x)# in terms of #x# yielding #3cos(3x)#.
This is
# f_x(x, y) = 3cos(3x) #
Similarly, for # f_y(x, y)#, we hold #x# fixed and differentiate with respect to #y# to get
This is
# f_y(x, y) = -5sin(5y) #