1.

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 – 2xsin⁡ 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^2I 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»

The CORRECT choice is (b) 3x^2 cos⁡ x^2 cos⁡ x^3 – 2xsin⁡ 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


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