NCERT Class 12 · Ch 5 Exercise 5.2 · Q6
NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.2 Question 6
Differentiate \(\sin^{2}{\left(x^{5} \right)} \cos{\left(x^{3} \right)}\) with respect to x; on R.
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: \(x^{2} \left(10 x^{2} \cos{\left(x^{3} \right)} \cos{\left(x^{5} \right)} - 3 \sin{\left(x^{3} \right)} \sin{\left(x^{5} \right)}\right) \sin{\left(x^{5} \right)}\)
- Work on R.
- Product rule: differentiate each variable factor once, keeping the other factors unchanged. Here \(\frac{d}{d x} \sin^{2}{\left(x^{5} \right)} \cos{\left(x^{3} \right)}\) = \(10 x^{4} \sin{\left(x^{5} \right)} \cos{\left(x^{3} \right)} \cos{\left(x^{5} \right)} - 3 x^{2} \sin{\left(x^{3} \right)} \sin^{2}{\left(x^{5} \right)}\).
- Combining the terms gives y′ = \(x^{2} \left(10 x^{2} \cos{\left(x^{3} \right)} \cos{\left(x^{5} \right)} - 3 \sin{\left(x^{3} \right)} \sin{\left(x^{5} \right)}\right) \sin{\left(x^{5} \right)}\).
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