Differentiate the following functions with respect to x:

(x sin x + cos x)(e^x + x² log x)

Let, y = (x sin x + cos x)( e^x + x² log 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 = (e^x + x² log 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 = e^x + x² log x

Using algebra of derivatives –

⇒

∵

∴ {using product rule}

⇒ …equation 3

∴ from equation 1, we can find dy/dx

∴

using equation 2 & 3, we get –

⇒

⇒

Hence,