Question

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.

(3 sec x – 4 cosec x) ( – 2 sin x + 5 cos x)

Answer 1

Let, y = (3 sec x – 4 cosec x)( – 2 sin x + 5 cos x)

⇒ y = – 6 sec x sin x + 15 sec x cos x + 8 sin x cosec x – 20cosec x cos x

⇒ y = – 6tan x + 15 + 8 – 20 cot x {∵ tan x cos x = sin x}

⇒ y = – 6 tan x – 20 cot x + 23

Differentiating y w.r.t x –

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

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 = (3 sec x – 4 cosec x) and v = ( – 2 sin x + 5 cos 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 –

∴ from equation 2, we can find dy/dx

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

Hence, Results are verified