NCERT Class 12 · Ch 5 Exercise 5.4 · Q5
NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.4 Question 5
Differentiate \(\log{\left(\cos{\left(e^{x} \right)} \right)}\) with respect to x; where cos(eˣ)>0.
💡 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: \(- e^{x} \tan{\left(e^{x} \right)}\)
- Work where cos(eˣ)>0.
- Differentiate the outer function and multiply by the inner derivative: \(\frac{d}{d x} \log{\left(\cos{\left(e^{x} \right)} \right)}\) = \(- \frac{e^{x} \sin{\left(e^{x} \right)}}{\cos{\left(e^{x} \right)}}\).
- Combining the terms gives y′ = \(- e^{x} \tan{\left(e^{x} \right)}\).
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