Step-by-step explanation:
Derivation using Product rule : -
To find the derivative of f(x) = sin 2x by the product rule, we have to express sin 2x as the product of two functions. Using the double angle formula of sin, sin 2x = 2 sin x cos x. Let us assume that u = 2 sin x and v = cos x. Then u' = 2 cos x and v' = -sin x. By product rule,
f '(x) = uv' + vu'
= (2 sin x) (- sin x) + (cos x) (2 cos x)
= 2 (cos2x - sin2x)
= 2 cos 2x
This is because, by the double angle formula of cos, cos 2x = cos2x - sin2x.
Thus, derivation of sin 2x has been found by using the product rule.
