NCERT Class 12 · Ch 5 Exercise 5.4 · Q9
NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.4 Question 9
Differentiate \(\frac{\cos{\left(x \right)}}{\log{\left(x \right)}}\) with respect to x; for x>0, x≠1.
💡 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: \(- \frac{\sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x \log{\left(x \right)}}}{\log{\left(x \right)}}\)
- Work for x>0, x≠1.
- Product rule: differentiate each variable factor once, keeping the other factors unchanged. Here \(\frac{d}{d x} \frac{\cos{\left(x \right)}}{\log{\left(x \right)}}\) = \(- \frac{\sin{\left(x \right)}}{\log{\left(x \right)}} - \frac{\cos{\left(x \right)}}{x \log{\left(x \right)}^{2}}\).
- Combining the terms gives y′ = \(- \frac{\sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x \log{\left(x \right)}}}{\log{\left(x \right)}}\).
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