NCERT Class 12 · Ch 5 Exercise 5.7 · Q8
NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.7 Question 8
Find the second derivative of \(\operatorname{atan}{\left(x \right)}\) 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: \(- \frac{2 x}{\left(x^{2} + 1\right)^{2}}\)
- Work on R.
- Differentiate the outer function and multiply by the inner derivative: \(\frac{d}{d x} \operatorname{atan}{\left(x \right)}\) = \(\frac{1}{x^{2} + 1}\).
- Differentiate the first derivative \(\frac{1}{x^{2} + 1}\) again.
- Power and chain rules: \(\frac{d}{d x} \frac{1}{x^{2} + 1}\) = \(- \frac{1}{\left(x^{2} + 1\right)^{2}}\) × \(2 x\) = \(- \frac{2 x}{\left(x^{2} + 1\right)^{2}}\).
- Combining the terms gives y″ = \(- \frac{2 x}{\left(x^{2} + 1\right)^{2}}\).
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