Differentiate each of the following functions by the product by the product rule and the other method and verify that answer from both the methods is the same.

(3x² + 2)²

Let, y = (3x² + 2)² = (3x² + 2)(3x² + 2)

⇒ y = 9x⁴ + 6x² + 6x² + 4

⇒ y = 9x⁴ + 12x² + 4

Differentiating y w.r.t x –

Using algebra of derivatives, we have –

Use formula of derivative of above function to get the result.

⇒ {∵

∴ …equation 1

Derivative using product rule –

We have to find dy/dx

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

u = (3x² + 2) and v = (3x² + 2)

∴ 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 2

As, u = (3x² + 2)

∴

⇒

⇒ …..equation 3 {∵ }

As, v = (3x² + 2)

⇒

⇒ …equation 4 {∵ }

∴ from equation 2, we can find dy/dx

∴

using equation 3 & 4, we get –

⇒

⇒

Hence,

….equation 5

Clearly from equation 1 and 5 we observed that both equations gave identical results.

Hence, Results are verified