Differentiate the following functions with respect to x:
(x sin x + cos x)(x cos x – sin x)
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,
