Find the derivatives :
i.) x sin x
ii.) sin x cos x
i.) To find the derivative of x sin x, we can use the product rule. Let u = x and v = sin x. Therefore,
u' = 1 (derivative of x)
v' = cos x (derivative of sin x)
The derivative of x sin x is given by:
(x sin x)' = u'v + uv'
= (1)(sin x) + (x)(cos x)
= sin x + x cos x
Therefore, the derivative of x sin x is sin x + x cos x.
ii.) To find the derivative of sin x cos x, we can use the product rule. Let u = sin x and v = cos x. Therefore,
u' = cos x (derivative of sin x)
v' = -sin x (derivative of cos x)
The derivative of sin x cos x is given by:
(sin x cos x)' = u'v + uv'
= (cos x)(cos x) + (sin x)(-sin x)
= cos^2 x - sin^2 x
= cos(2x)
Therefore, the derivative of sin x cos x is cos(2x).