Let, y = (x sin x + cos x)(x cos x – sin x)

We have to find dy/dx

As we can observe that y is a product of two functions say u and v where,

u = x sin x + cos x and v = x cos x – sin x

∴ y = uv

As we know that to find the derivative of product of two function we apply product rule of differentiation.

By product rule, we have –

…equation 1

As, u = x sin x + cos x

∴

Using algebra of derivatives –

⇒

∵

∴ {using product rule}

⇒ …equation 2

As, v = x cos x – sin x

Using algebra of derivatives –

⇒

∵

∴ {using product rule}

⇒ …equation 3

∴ from equation 1, we can find dy/dx

∴

using equation 2 & 3, we get –

⇒

⇒

As, we know that: cos²x – sin²x = cos 2x & 2sin x cos x = sin 2x

Hence,