Frankly, by the time a person is studing three dimensional problems and parametric equations, differentiating trig functions should be easy. What was giving you difficulty here? And why in the world is this posted under "Differential Equations"?
By the way, I would not multiply the first one:
f(x)= (4+ sin(20t))cos(t) so f'= (4+ sin(20t))'cos(t)+ (4+ sin(20t)(cos(t))'
= (20cos(20t))cos(t)- (4+ sin(20t)sin(t).