Differentiate in two ways, using product rule and otherwise, the function

(1 + 2tan x)(5 + 4 cos x). Verify that the answers are the same.

Let, y = (1 + 2 tan x)(5 + 4 cos x)

⇒ y = 5 + 4 cos x + 10 tan x + 8 tan x cos x

⇒ y = 5 + 4 cos x + 10 tan x + 8 sin x {∵ tan x cos x = sin x}

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 = (1 + 2tan x) and v = (5 + 4cos 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 2

As, u = (1 + 2tan x)

∴

⇒

⇒ …..equation 3 {∵ }

As, v = 5 + 4cos x

⇒

⇒ …equation 4 {∵ }

∴ from equation 2, we can find dy/dx

∴

using equation 3 & 4, we get –

⇒

⇒

∵ sin x = tan x cos x , so we get –

⇒

⇒

⇒ [∵ 1 – sin² x = cos² x ]

∴

Hence,

….equation 5

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

Hence, Results are verified