Question

What is the value of \(\frac{d}{dx}\) (sin⁡ x^3 cos⁡ x^2)?

(a) 3x^2 cos x^2 cos⁡ x^3 + 2x sin⁡ x^3 sin x^2

(b) 3x^2 cos⁡ x^2 cos⁡ x^3 – 2x  sin⁡ x^3 sin x^2

(c) 2x cos x^2 cos⁡ x^3 – 2x sin⁡ x^3 sin x^2

(d) 2x cos x^2 cos⁡ x^3 + 3x^2 sin⁡ x^3 sin x^2

I have been asked this question in an interview.

The origin of the question is Derivatives topic in chapter Limits and Derivatives of Mathematics – Class 11

Answer 1

The correct choice is (b) 3x^2 cos⁡ x^2 cos⁡ x^3 – 2x  sin⁡ x^3 sin x^2

Best explanation: We follow product rule\(\frac{d}{dx}\) (f.g)= g.\(\frac{d}{dx}\) (f)+ f.\(\frac{dy}{dx}\) (g)

Here f = sin⁡ x^3 and g = cos⁡ x^2

\(\frac{d}{dx}\) (f) = 3x^2 cos⁡ x^3

\(\frac{d}{dx}\) (g) = -2x sin x^2

We now substitute this in our main equation,

= cos⁡ x^2.3x^2 cos⁡ x^3 + sin⁡ x^3.(-2x sin x^2)

= 3x^2 cos x^2 cos⁡ x^3 – 2x sin⁡ x^3 sin x^2