NCERT Class 12 · Ch 5 Exercise 5.4 · Q6
NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.4 Question 6
Differentiate \(e^{x} + e^{x^{2}} + e^{x^{3}} + e^{x^{4}} + e^{x^{5}}\) 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: \(5 x^{4} e^{x^{5}} + 4 x^{3} e^{x^{4}} + 3 x^{2} e^{x^{3}} + 2 x e^{x^{2}} + e^{x}\)
- Work on R.
- Differentiate the outer function and multiply by the inner derivative: \(\frac{d}{d x} e^{x}\) = \(e^{x}\).
- Differentiate the outer function and multiply by the inner derivative: \(\frac{d}{d x} e^{x^{2}}\) = \(2 x e^{x^{2}}\).
- Differentiate the outer function and multiply by the inner derivative: \(\frac{d}{d x} e^{x^{3}}\) = \(3 x^{2} e^{x^{3}}\).
- Differentiate the outer function and multiply by the inner derivative: \(\frac{d}{d x} e^{x^{4}}\) = \(4 x^{3} e^{x^{4}}\).
- Differentiate the outer function and multiply by the inner derivative: \(\frac{d}{d x} e^{x^{5}}\) = \(5 x^{4} e^{x^{5}}\).
- Sum rule: \(\frac{d}{d x} \left(e^{x} + e^{x^{2}} + e^{x^{3}} + e^{x^{4}} + e^{x^{5}}\right)\) = \(5 x^{4} e^{x^{5}} + 4 x^{3} e^{x^{4}} + 3 x^{2} e^{x^{3}} + 2 x e^{x^{2}} + e^{x}\).
- Combining the terms gives y′ = \(5 x^{4} e^{x^{5}} + 4 x^{3} e^{x^{4}} + 3 x^{2} e^{x^{3}} + 2 x e^{x^{2}} + e^{x}\).
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