NCERT Class 12 · Ch 5 Exercise 5.4 · Q1

NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.4 Question 1

Differentiate \(\frac{e^{x}}{\sin{\left(x \right)}}\) with respect to x; where sin x≠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.

Final Answer: \(\frac{\left(1 - \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right) e^{x}}{\sin{\left(x \right)}}\)

Work where sin x≠0.
Product rule: differentiate each variable factor once, keeping the other factors unchanged. Here \(\frac{d}{d x} \frac{e^{x}}{\sin{\left(x \right)}}\) = \(\frac{e^{x}}{\sin{\left(x \right)}} - \frac{e^{x} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}\).
Combining the terms gives y′ = \(\frac{\left(1 - \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right) e^{x}}{\sin{\left(x \right)}}\).

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