NCERT Class 12 · Ch 5 Exercise 5.7 · Q3
NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.7 Question 3
Find the second derivative of \(x \cos{\left(x \right)}\) 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: \(- (x \cos{\left(x \right)} + 2 \sin{\left(x \right)})\)
- Work on R.
- Product rule: differentiate each variable factor once, keeping the other factors unchanged. Here \(\frac{d}{d x} x \cos{\left(x \right)}\) = \(- x \sin{\left(x \right)} + \cos{\left(x \right)}\).
- Differentiate the first derivative \(- x \sin{\left(x \right)} + \cos{\left(x \right)}\) again.
- Product rule: differentiate each variable factor once, keeping the other factors unchanged. Here \(\frac{d}{d x} \left(- x \sin{\left(x \right)}\right)\) = \(- x \cos{\left(x \right)} - \sin{\left(x \right)}\).
- Differentiate the outer function and multiply by the inner derivative: \(\frac{d}{d x} \cos{\left(x \right)}\) = \(- \sin{\left(x \right)}\).
- Sum rule: \(\frac{d}{d x} \left(- x \sin{\left(x \right)} + \cos{\left(x \right)}\right)\) = \(- x \cos{\left(x \right)} - 2 \sin{\left(x \right)}\).
- Combining the terms gives y″ = \(- (x \cos{\left(x \right)} + 2 \sin{\left(x \right)})\).
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