NCERT Class 12 · Ch 5 Exercise 5.4 · Q4
NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.4 Question 4
Differentiate \(\frac{e^{- x}}{\sqrt{1 + e^{- 2 x}}}\) 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: \(- \frac{e^{2 x}}{\left(e^{2 x} + 1\right)^{\frac{3}{2}}}\)
- Work on R.
- Product rule: differentiate each variable factor once, keeping the other factors unchanged. Here \(\frac{d}{d x} \frac{e^{- x}}{\sqrt{1 + e^{- 2 x}}}\) = \(- \frac{e^{- x}}{\sqrt{1 + e^{- 2 x}}} + \frac{e^{- 3 x}}{\left(1 + e^{- 2 x}\right)^{\frac{3}{2}}}\).
- Combining the terms gives y′ = \(- \frac{e^{2 x}}{\left(e^{2 x} + 1\right)^{\frac{3}{2}}}\).
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