We have: #sin^(2)(8 x) - (pi) x#
#= (sin(8 x))^(2) - pi x#
First, let's apply the difference rule:
#Rightarrow frac(d)(dx) ((sin(8 x))^(2) - pi x) = frac(d)(dx)((sin(8 x))^(2)) - frac(d)(dx)(pi x)#
#Rightarrow frac(d)(dx) ((sin(8 x))^(2) - pi x) = frac(d)(dx) ((sin(8 x))^(2)) - pi#
Then, let's use the chain rule.
Let #u = sin(8 x) Rightarrow u' = cos(8 x)#, #v = u^(2) Rightarrow v' = 2 u# and #w = 8 x Rightarrow w' = 8#:
#Rightarrow frac(d)(dx) ((sin(8 x))^(2) - pi x) = u' cdot v' cdot w' - pi#
#Rightarrow frac(d)(dx) ((sin(8 x))^(2) - pi x) = cos(8 x) cdot 2 u cdot 8 - pi#
#Rightarrow frac(d)(dx) ((sin(8 x))^(2) - pi x) = 8 cdot 2 u cos(8 x) - pi#
Now, let's replace #u# with #sin(8 x)#:
#Rightarrow frac(d)(dx) ((sin(8 x))^(2) - pi x) = 8 cdot 2 sin(8 x) cos(8 x) - pi#
Let's apply the double angle formula for #sin(x)#; #sin(2 x) = 2 sin(x) cos(x)#:
#Rightarrow frac(d)(dx) ((sin(8 x))^(2) - pi x) = 8 cdot sin(2 cdot 8 x) - pi#
#therefore frac(d)(dx) ((sin(8 x))^(2) - pi x) = 8 sin(16 x) - pi#