What is derivative of f(x)= (sin2x+cos2x)(sin2x-cos2x)?
(a-b)(a+b) =`a^2 - b^2 `Let a = sin x and b = cos x
(sin 2x + cos 2x)(sin 2x - cos 2x) = ` sin^2 (2x) - cos^2 (2x) `We'll factor -1 and we'll get:
(sin 2x + cos 2x)(sin 2x - cos 2x) = -(` cos^2 (2x) - sin^2 (2x)` )This difference of two squares represents the formula for the double angle:
-(`cos^2 (2x) - sin^2 (2x)` ) = - cos (4x)We'll apply the chain rule to differentiate with respect to x:
f'(x) = -(-sin 4x)*(4x)'f'(x) = 4sin 4x
The requested derivative of the given function is
f'(x) = 4sin 4x.