I believe you mean #f(x) = sin(x) cdot cos(2 x)#
We can differentiate this function using the chain rule and the product rule.
Let #u = 2 x Rightarrow u' = 2# and #v = cos(u) Rightarrow v' = - sin(u)#:
#Rightarrow f'(x) = frac(d)(dx)(sin(x)) cdot (cos(2 x)) + frac(d)(dx)(cos(2 x)) cdot (sin(x))#
#Rightarrow f'(x) = cos(x) cdot cos(2 x) + u' cdot v' cdot sin(x)#
#Rightarrow f'(x) = cos(x) cos(2 x) + 2 cdot - sin(u) cdot sin(x)#
#Rightarrow f'(x) = cos(x) cos(2 x) - 2 sin(u) sin(x)#
Let's replace #u# with #2 x#:
#Rightarrow f'(x) = cos(x) cos(2 x) - 2 sin(2 x) sin(x)#
#therefore f'(x) = cos(x) cos(2 x) - 2 sin(x) sin(2 x)#
