TGT PGT OBJECTIVE MATHS DIFFRENTION QUESTION

Below are 50 TGT (Trained Graduate Teacher) level objective mathematics questions on the topic of **Calculus: Differentiation**. These questions are designed to be concise, cover a range of concepts in differentiation, and align with the level expected for TGT competitive exams. Each question includes four options, the correct answer, and a brief explanation. — ### **TGT Level Objective Questions on Differentiation** 1. **Question:** What is the derivative of \( f(x) = 5x^3 + 2x^2 – 3x + 4 \) with respect to \( x \)? A) \( 15x^2 + 4x – 3 \) B) \( 15x^2 + 4x + 3 \) C) \( 5x^2 + 2x – 3 \) D) \( 15x^2 – 4x – 3 \) **Answer:** A) \( 15x^2 + 4x – 3 \) **Explanation:** Differentiate term by term: \( \frac{d}{dx}(5x^3) = 15x^2 \), \( \frac{d}{dx}(2x^2) = 4x \), \( \frac{d}{dx}(-3x) = -3 \), \( \frac{d}{dx}(4) = 0 \). 2. **Question:** If \( y = \sin(3x) \), what is \( \frac{dy}{dx} \)? A) \( 3\cos(3x) \) B) \( \cos(3x) \) C) \( -\sin(3x) \) D) \( -3\cos(3x) \) **Answer:** A) \( 3\cos(3x) \) **Explanation:** Using the chain rule, \( \frac{d}{dx}[\sin(3x)] = \cos(3x) \cdot \frac{d}{dx}(3x) = \cos(3x) \cdot 3 \). 3. **Question:** The derivative of \( f(x) = e^{2x} \) is: A) \( e^{2x} \) B) \( 2e^{2x} \) C) \( e^x \) D) \( 2e^x \) **Answer:** B) \( 2e^{2x} \) **Explanation:** \( \frac{d}{dx}(e^{2x}) = e^{2x} \cdot \frac{d}{dx}(2x) = e^{2x} \cdot 2 \). 4. **Question:** If \( y = \ln(4x) \), then \( \frac{dy}{dx} \) is: A) \( \frac{1}{x} \) B) \( \frac{4}{x} \) C) \( \frac{1}{4x} \) D) \( 4x \) **Answer:** C) \( \frac{1}{4x} \) **Explanation:** \( \frac{d}{dx}[\ln(4x)] = \frac{1}{4x} \cdot \frac{d}{dx}(4x) = \frac{1}{4x} \cdot 4 = \frac{1}{x} \). 5. **Question:** What is the derivative of \( f(x) = x^2 \cos(x) \)? A) \( 2x \cos(x) + x^2 \sin(x) \) B) \( 2x \cos(x) – x^2 \sin(x) \) C) \( x^2 \sin(x) – 2x \cos(x) \) D) \( 2x \sin(x) + x^2 \cos(x) \) **Answer:** B) \( 2x \cos(x) – x^2 \sin(x) \) **Explanation:** Use the product rule: \( \frac{d}{dx}(x^2 \cos(x)) = 2x \cos(x) + x^2 (-\sin(x)) \). 6. **Question:** If \( y = \frac{x^2}{x + 1} \), what is \( \frac{dy}{dx} \)? A) \( \frac{x^2 + 2x}{(x + 1)^2} \) B) \( \frac{x^2 – 2x}{(x + 1)^2} \) C) \( \frac{2x}{(x + 1)^2} \) D) \( \frac{x^2}{(x + 1)^2} \) **Answer:** A) \( \frac{x^2 + 2x}{(x + 1)^2} \) **Explanation:** Use the quotient rule: \( \frac{d}{dx}\left(\frac{x^2}{x + 1}\right) = \frac{(2x)(x + 1) – x^2 \cdot 1}{(x + 1)^2} = \frac{2x^2 + 2x – x^2}{(x + 1)^2} = \frac{x^2 + 2x}{(x + 1)^2} \). 7. **Question:** The derivative of \( f(x) = \tan(x) \) is: A) \( \sec^2(x) \) B) \( \cos^2(x) \) C) \( -\sec^2(x) \) D) \( \sin^2(x) \) **Answer:** A) \( \sec^2(x) \) **Explanation:** \( \frac{d}{dx}(\tan(x)) = \sec^2(x) \). 8. **Question:** If \( y = x \ln(x) \), then \( \frac{dy}{dx} \) is: A) \( \ln(x) + 1 \) B) \( \ln(x) – 1 \) C) \( x \ln(x) \) D) \( \frac{1}{x} \) **Answer:** A) \( \ln(x) + 1 \) **Explanation:** Use the product rule: \( \frac{d}{dx}(x \ln(x)) = \ln(x) \cdot 1 + x \cdot \frac{1}{x} = \ln(x) + 1 \). 9. **Question:** What is the derivative of \( f(x) = \sqrt{x} \)? A) \( \frac{1}{2\sqrt{x}} \) B) \( \frac{1}{\sqrt{x}} \) C) \( \sqrt{x} \) D) \( \frac{1}{x} \) **Answer:** A) \( \frac{1}{2\sqrt{x}} \) **Explanation:** Rewrite \( \sqrt{x} = x^{1/2} \), then \( \frac{d}{dx}(x^{1/2}) = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}} \). 10. **Question:** If \( y = e^x \sin(x) \), then \( \frac{dy}{dx} \) is: A) \( e^x \cos(x) \) B) \( e^x [\sin(x) + \cos(x)] \) C) \( e^x [\sin(x) – \cos(x)] \) D) \( e^x \cos(x) + \sin(x) \) **Answer:** B) \( e^x [\sin(x) + \cos(x)] \) **Explanation:** Use the product rule: \( \frac{d}{dx}(e^x \sin(x)) = e^x \sin(x) + e^x \cos(x) = e^x [\sin(x) + \cos(x)] \). 11. **Question:** The derivative of \( f(x) = \frac{1}{x^2} \) is: A) \( -\frac{2}{x^3} \) B) \( \frac{2}{x^3} \) C) \( -\frac{1}{x^3} \) D) \( \frac{1}{x^3} \) **Answer:** A) \( -\frac{2}{x^3} \) **Explanation:** Rewrite \( \frac{1}{x^2} = x^{-2} \), then \( \frac{d}{dx}(x^{-2}) = -2x^{-3} = -\frac{2}{x^3} \). 12. **Question:** If \( y = \cos(2x) \), what is \( \frac{dy}{dx} \)? A) \( 2\sin(2x) \) B) \( -2\sin(2x) \) C) \( \sin(2x) \) D) \( -\sin(2x) \) **Answer:** B) \( -2\sin(2x) \) **Explanation:** \( \frac{d}{dx}[\cos(2x)] = -\sin(2x) \cdot 2 = -2\sin(2x) \). 13. **Question:** The derivative of \( f(x) = x^3 e^x \) is: A) \( x^3 e^x + 3x^2 e^x \) B) \( x^3 e^x – 3x^2 e^x \) C) \( 3x^2 e^x \) D) \( x^3 e^x \) **Answer:** A) \( x^3 e^x + 3x^2 e^x \) **Explanation:** Use the product rule: \( \frac{d}{dx}(x^3 e^x) = 3x^2 e^x + x^3 e^x \). 14. **Question:** If \( y = \frac{\sin(x)}{x} \), what is \( \frac{dy}{dx} \)? A) \( \frac{\cos(x)}{x} – \frac{\sin(x)}{x^2} \) B) \( \frac{\sin(x)}{x^2} + \cos(x) \) C) \( \frac{\cos(x)}{x} + \frac{\sin(x)}{x^2} \) D) \( \frac{\sin(x)}{x} \) **Answer:** A) \( \frac{\cos(x)}{x} – \frac{\sin(x)}{x^2} \) **Explanation:** Use the quotient rule: \( \frac{d}{dx}\left(\frac{\sin(x)}{x}\right) = \frac{\cos(x) \cdot x – \sin(x) \cdot 1}{x^2} \). 15. **Question:** What is the derivative of \( f(x) = \ln(x^2 + 1) \)? A) \( \frac{2x}{x^2 + 1} \) B) \( \frac{x}{x^2 + 1} \) C) \( \frac{1}{x^2 + 1} \) D) \( \frac{2x}{x^2} \) **Answer:** A) \( \frac{2x}{x^2 + 1} \) **Explanation:** \( \frac{d}{dx}[\ln(x^2 + 1)] = \frac{1}{x^2 + 1} \cdot 2x \). 16. **Question:** If \( y = x^2 + \frac{1}{x^2} \), then \( \frac{dy}{dx} \) is: A) \( 2x – \frac{2}{x^3} \) B) \( 2x + \frac{2}{x^3} \) C) \( 2x – \frac{1}{x^3} \) D) \( x – \frac{1}{x^3} \) **Answer:** A) \( 2x – \frac{2}{x^3} \) **Explanation:** Differentiate term by term: \( \frac{d}{dx}(x^2) = 2x \), \( \frac{d}{dx}(x^{-2}) = -2x^{-3} \). 17. **Question:** The derivative of \( f(x) = \sec(x) \) is: A) \( \sec(x) \tan(x) \) B) \( -\sec(x) \tan(x) \) C) \( \sec^2(x) \) D) \( \tan^2(x) \) **Answer:** A) \( \sec(x) \tan(x) \) **Explanation:** \( \frac{d}{dx}(\sec(x)) = \sec(x) \tan(x) \). 18. **Question:** If \( y = e^{x^2} \), what is \( \frac{dy}{dx} \)? A) \( e^{x^2} \) B) \( 2x e^{x^2} \) C) \( x e^{x^2} \) D) \( e^x \) **Answer:** B) \( 2x e^{x^2} \) **Explanation:** \( \frac{d}{dx}(e^{x^2}) = e^{x^2} \cdot 2x \). 19. **Question:** The derivative of \( f(x) = \sin^2(x) \) is: A) \( 2\sin(x) \cos(x) \) B) \( \sin(x) \cos(x) \) C) \( 2\cos^2(x) \) D) \( -\sin^2(x) \) **Answer:** A) \( 2\sin(x) \cos(x) \) **Explanation:** Rewrite \( \sin^2(x) = (\sin(x))^2 \), then use the chain rule: \( 2\sin(x) \cos(x) \). 20. **Question:** If \( y = \frac{1}{\sqrt{x}} \), what is \( \frac{dy}{dx} \)? A) \( -\frac{1}{2x^{3/2}} \) B) \( \frac{1}{2x^{3/2}} \) C) \( -\frac{1}{2\sqrt{x}} \) D) \( \frac{1}{2\sqrt{x}} \) **Answer:** A) \( -\frac{1}{2x^{3/2}} \) **Explanation:** Rewrite \( \frac{1}{\sqrt{x}} = x^{-1/2} \), then \( \frac{d}{dx}(x^{-1/2}) = -\frac{1}{2} x^{-3/2} \). 21. **Question:** The derivative of \( f(x) = x \tan(x) \) is: A) \( \tan(x) + x \sec^2(x) \) B) \( \tan(x) – x \sec^2(x) \) C) \( x \tan(x) + \sec^2(x) \) D) \( \sec^2(x) – \tan(x) \) **Answer:** A) \( \tan(x) + x \sec^2(x) \) **Explanation:** Use the product rule: \( \frac{d}{dx}(x \tan(x)) = \tan(x) + x \sec^2(x) \). 22. **Question:** If \( y = \ln(\cos(x)) \), then \( \frac{dy}{dx} \) is: A) \( -\tan(x) \) B) \( \tan(x) \) C) \( -\sec(x) \) D) \( \sec(x) \) **Answer:** A) \( -\tan(x) \) **Explanation:** \( \frac{d}{dx}[\ln(\cos(x))] = \frac{1}{\cos(x)} \cdot (-\sin(x)) = -\tan(x) \). 23. **Question:** What is the derivative of \( f(x) = x^2 e^{-x} \)? A) \( x^2 e^{-x} – 2x e^{-x} \) B) \( 2x e^{-x} – x^2 e^{-x} \) C) \( x^2 e^{-x} + 2x e^{-x} \) D) \( 2x e^{-x} + x^2 e^{-x} \) **Answer:** B) \( 2x e^{-x} – x^2 e^{-x} \) **Explanation:** Use the product rule: \( \frac{d}{dx}(x^2 e^{-x}) = 2x e^{-x} + x^2 (-e^{-x}) \). 24. **Question:** If \( y = \frac{x}{\sin(x)} \), then \( \frac{dy}{dx} \) is: A) \( \frac{\sin(x) – x \cos(x)}{\sin^2(x)} \) B) \( \frac{\sin(x) + x \cos(x)}{\sin^2(x)} \) C) \( \frac{x \cos(x)}{\sin^2(x)} \) D) \( \frac{\cos(x)}{\sin(x)} \) **Answer:** A) \( \frac{\sin(x) – x \cos(x)}{\sin^2(x)} \) **Explanation:** Use the quotient rule: \( \frac{d}{dx}\left(\frac{x}{\sin(x)}\right) = \frac{1 \cdot \sin(x) – x \cos(x)}{\sin^2(x)} \). 25. **Question:** The derivative of \( f(x) = \cot(x) \) is: A) \( -\csc^2(x) \) B) \( \csc^2(x) \) C) \( -\sec^2(x) \) D) \( \sec^2(x) \) **Answer:** A) \( -\csc^2(x) \) **Explanation:** \( \frac{d}{dx}(\cot(x)) = -\csc^2(x) \). 26. **Question:** If \( y = e^{3x} \cos(x) \), then \( \frac{dy}{dx} \) is: A) \( e^{3x} [3\cos(x) – \sin(x)] \) B) \( e^{3x} [3\cos(x) + \sin(x)] \) C) \( e^{3x} [\cos(x) – 3\sin(x)] \) D) \( e^{3x} [\cos(x) + 3\sin(x)] \) **Answer:** A) \( e^{3x} [3\cos(x) – \sin(x)] \) **Explanation:** Use the product rule: \( \frac{d}{dx}(e^{3x} \cos(x)) = 3e^{3x} \cos(x) + e^{3x} (-\sin(x)) \). 27. **Question:** What is the derivative of \( f(x) = \ln(x^3) \)? A) \( \frac{3}{x} \) B) \( \frac{1}{x^3} \) C) \( \frac{1}{x} \) D) \( \frac{3}{x^3} \) **Answer:** A) \( \frac{3}{x} \) **Explanation:** \( \ln(x^3) = 3\ln(x) \), so \( \frac{d}{dx}(3\ln(x)) = 3 \cdot \frac{1}{x} \). 28. **Question:** If \( y = x^2 \sin(2x) \), then \( \frac{dy}{dx} \) is: A) \( 2x \sin(2x) + 2x^2 \cos(2x) \) B) \( 2x \sin(2x) – 2x^2 \cos(2x) \) C) \( x^2 \cos(2x) \) D) \( 2x \cos(2x) \) **Answer:** A) \( 2x \sin(2x) + 2x^2 \cos(2x) \) **Explanation:** Use the product rule: \( \frac{d}{dx}(x^2 \sin(2x)) = 2x \sin(2x) + x^2 \cdot 2\cos(2x) \). 29. **Question:** The derivative of \( f(x) = \frac{1}{x + 1} \) is: A) \( -\frac{1}{(x + 1)^2} \) B) \( \frac{1}{(x + 1)^2} \) C) \( -\frac{1}{x + 1} \) D) \( \frac{1}{x + 1} \) **Answer:** A) \( -\frac{1}{(x + 1)^2} \) **Explanation:** Rewrite \( \frac{1}{x + 1} = (x + 1)^{-1} \), then \( \frac{d}{dx}((x + 1)^{-1}) = -1 (x + 1)^{-2} \). 30. **Question:** If \( y = \sin(x^2) \), what is \( \frac{dy}{dx} \)? A) \( 2x \cos(x^2) \) B) \( \cos(x^2) \) C) \( x \cos(x^2) \) D) \( 2x \sin(x^2) \) **Answer:** A) \( 2x \cos(x^2) \) **Explanation:** Use the chain rule: \( \frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x \). 31. **Question:** The derivative of \( f(x) = e^x + x e^x \) is: A) \( e^x + x e^x \) B) \( e^x + e^x + x e^x \) C) \( e^x – x e^x \) D) \( x e^x \) **Answer:** B) \( e^x + e^x + x e^x \) **Explanation:** Differentiate term by term: \( \frac{d}{dx}(e^x) = e^x \), \( \frac{d}{dx}(x e^x) = e^x + x e^x \). 32. **Question:** If \( y = \tan^2(x) \), then \( \frac{dy}{dx} \) is: A) \( 2\tan(x) \sec^2(x) \) B) \( \tan(x) \sec^2(x) \) C) \( 2\sec^2(x) \) D) \( \sec^2(x) \) **Answer:** A) \( 2\tan(x) \sec^2(x) \) **Explanation:** Rewrite \( \tan^2(x) = (\tan(x))^2 \), then use the chain rule: \( 2\tan(x) \cdot \sec^2(x) \). 33. **Question:** What is the derivative of \( f(x) = \frac{x^2 + 1}{x} \)? A) \( 1 – \frac{1}{x^2} \) B) \( 1 + \frac{1}{x^2} \) C) \( 2x – \frac{1}{x^2} \) D) \( 2x + \frac{1}{x^2} \) **Answer:** A) \( 1 – \frac{1}{x^2} \) **Explanation:** Rewrite \( \frac{x^2 + 1}{x} = x + \frac{1}{x} \), then \( \frac{d}{dx}(x + x^{-1}) = 1 – x^{-2} \). 34. **Question:** If \( y = x^3 \ln(x) \), then \( \frac{dy}{dx} \) is: A) \( 3x^2 \ln(x) + x^2 \) B) \( 3x^2 \ln(x) – x^2 \) C) \( x^3 \ln(x) + x^2 \) D) \( 3x^2 + \ln(x) \) **Answer:** A) \( 3x^2 \ln(x) + x^2 \) **Explanation:** Use the product rule: \( \frac{d}{dx}(x^3 \ln(x)) = 3x^2 \ln(x) + x^3 \cdot \frac{1}{x} \). 35. **Question:** The derivative of \( f(x) = \csc(x) \) is: A) \( -\csc(x) \cot(x) \) B) \( \csc(x) \cot(x) \) C) \( -\sec(x) \tan(x) \) D) \( \sec(x) \tan(x) \) **Answer:** A) \( -\csc(x) \cot(x) \) **Explanation:** \( \frac{d}{dx}(\csc(x)) = -\csc(x) \cot(x) \). 36. **Question:** If \( y = \frac{e^x}{x} \), then \( \frac{dy}{dx} \) is: A) \( \frac{e^x (x – 1)}{x^2} \) B) \( \frac{e^x (x + 1)}{x^2} \) C) \( \frac{e^x}{x^2} \) D) \( \frac{e^x}{x} \) **Answer:** A) \( \frac{e^x (x – 1)}{x^2} \) **Explanation:** Use the quotient rule: \( \frac{d}{dx}\left(\frac{e^x}{x}\right) = \frac{e^x \cdot x – e^x \cdot 1}{x^2} \). 37. **Question:** What is the derivative of \( f(x) = x^2 + \sin(x) \)? A) \( 2x + \cos(x) \) B) \( 2x – \cos(x) \) C) \( x^2 + \cos(x) \) D) \( 2x + \sin(x) \) **Answer:** A) \( 2x + \cos(x) \) **Explanation:** Differentiate term by term: \( \frac{d}{dx}(x^2) = 2x \), \( \frac{d}{dx}(\sin(x)) = \cos(x) \). 38. **Question:** If \( y = \ln(1 + x^2) \), then \( \frac{dy}{dx} \) is: A) \( \frac{2x}{1 + x^2} \) B) \( \frac{x}{1 + x^2} \) C) \( \frac{1}{1 + x^2} \) D) \( \frac{2x}{x^2} \) **Answer:** A) \( \frac{2x}{1 + x^2} \) **Explanation:** \( \frac{d}{dx}[\ln(1 + x^2)] = \frac{1}{1 + x^2} \cdot 2x \). 39. **Question:** The derivative of \( f(x) = x e^{2x} \) is: A) \( e^{2x} + 2x e^{2x} \) B) \( e^{2x} – 2x e^{2x} \) C) \( 2x e^{2x} \) D) \( x e^{2x} \) **Answer:** A) \( e^{2x} + 2x e^{2x} \) **Explanation:** Use the product rule: \( \frac{d}{dx}(x e^{2x}) = e^{2x} + x \cdot 2e^{2x} \). 40. **Question:** If \( y = \cos^2(x) \), then \( \frac{dy}{dx} \) is: A) \( -2\sin(x) \cos(x) \) B) \( 2\sin(x) \cos(x) \) C) \( -\sin^2(x) \) D) \( \cos^2(x) \) **Answer:** A) \( -2\sin(x) \cos(x) \) **Explanation:** Rewrite \( \cos^2(x) = (\cos(x))^2 \), then use the chain rule: \( 2\cos(x) \cdot (-\sin(x)) \). 41. **Question:** What is the derivative of \( f(x) = \frac{1}{\ln(x)} \)? A) \( -\frac{1}{x \ln^2(x)} \) B) \( \frac{1}{x \ln^2(x)} \) C) \( -\frac{1}{\ln(x)} \) D) \( \frac{1}{\ln(x)} \) **Answer:** A) \( -\frac{1}{x \ln^2(x)} \) **Explanation:** Rewrite \( \frac{1}{\ln(x)} = (\ln(x))^{-1} \), then \( \frac{d}{dx}((\ln(x))^{-1}) = -1 (\ln(x))^{-2} \cdot \frac{1}{x} \). 42. **Question:** If \( y = x^2 \tan(x) \), then \( \frac{dy}{dx} \) is: A) \( 2x \tan(x) + x^2 \sec^2(x) \) B) \( 2x \tan(x) – x^2 \sec^2(x) \) C) \( x^2 \tan(x) + 2x \sec^2(x) \) D) \( 2x \sec^2(x) \) **Answer:** A) \( 2x \tan(x) + x^2 \sec^2(x) \) **Explanation:** Use the product rule: \( \frac{d}{dx}(x^2 \tan(x)) = 2x \tan(x) + x^2 \sec^2(x) \). 43. **Question:** The derivative of \( f(x) = \sin(x) \cos(x) \) is: A) \( \cos^2(x) – \sin^2(x) \) B) \( \sin^2(x) – \cos^2(x) \) C) \( \sin^2(x) + \cos^2(x) \) D) \( \cos^2(x) + \sin^2(x) \) **Answer:** A) \( \cos^2(x) – \sin^2(x) \) **Explanation:** Use the product rule: \( \frac{d}{dx}(\sin(x) \cos(x)) = \cos(x) \cos(x) + \sin(x) (-\sin(x)) \). 44. **Question:** If \( y = e^{-x} \sin(x) \), then \( \frac{dy}{dx} \) is: A) \( e^{-x} [\cos(x) – \sin(x)] \) B) \( e^{-x} [\sin(x) + \cos(x)] \) C) \( e^{-x} [\cos(x) + \sin(x)] \) D) \( e^{-x} [\sin(x) – \cos(x)] \) **Answer:** A) \( e^{-x} [\cos(x) – \sin(x)] \) **Explanation:** Use the product rule: \( \frac{d}{dx}(e^{-x} \sin(x)) = (-e^{-x}) \sin(x) + e^{-x} \cos(x) \). 45. **Question:** What is the derivative of \( f(x) = \frac{x^3}{x^2 + 1} \)? A) \( \frac{3x^2 (x^2 + 1) – 2x^4}{(x^2 + 1)^2} \) B) \( \frac{3x^2 (x^2 + 1) + 2x^4}{(x^2 + 1)^2} \) C) \( \frac{x^2 (x^2 + 1)}{(x^2 + 1)^2} \) D) \( \frac{3x^2}{(x^2 + 1)^2} \) **Answer:** A) \( \frac{3x^2 (x^2 + 1) – 2x^4}{(x^2 + 1)^2} \) **Explanation:** Use the quotient rule: \( \frac{d}{dx}\left(\frac{x^3}{x^2 + 1}\right) = \frac{3x^2 (x^2 + 1) – x^3 \cdot 2x}{(x^2 + 1)^2} \). 46. **Question:** If \( y = \ln(\sin(x)) \), then \( \frac{dy}{dx} \) is: A) \( \cot(x) \) B) \( -\cot(x) \) C) \( \csc(x) \) D) \( -\csc(x) \) **Answer:** A) \( \cot(x) \) **Explanation:** \( \frac{d}{dx}[\ln(\sin(x))] = \frac{1}{\sin(x)} \cdot \cos(x) = \cot(x) \). 47. **Question:** The derivative of \( f(x) = x^4 e^{-2x} \) is: A) \( 4x^3 e^{-2x} – 2x^4 e^{-2x} \) B) \( 4x^3 e^{-2x} + 2x^4 e^{-2x} \) C) \( x^4 e^{-2x} \) D) \( 4x^3 e^{-2x} \) **Answer:** A) \( 4x^3 e^{-2x} – 2x^4 e^{-2x} \) **Explanation:** Use the product rule: \( \frac{d}{dx}(x^4 e^{-2x}) = 4x^3 e^{-2x} + x^4 (-2e^{-2x}) \). 48. **Question:** If \( y = \sin(x) + \cos(x) \), then \( \frac{dy}{dx} \) is: A) \( \cos(x) – \sin(x) \) B) \( \sin(x) – \cos(x) \) C) \( \sin(x) + \cos(x) \) D) \( -\sin(x) – \cos(x) \) **Answer:** A) \( \cos(x) – \sin(x) \) **Explanation:** Differentiate term by term: \( \frac{d}{dx}(\sin(x)) = \cos(x) \), \( \frac{d}{dx}(\cos(x)) = -\sin(x) \). 49. **Question:** What is the derivative of \( f(x) = \frac{\cos(x)}{x^2} \)? A) \( \frac{-\sin(x) x^2 – 2x \cos(x)}{x^4} \) B) \( \frac{\sin(x) x^2 – 2x \cos(x)}{x^4} \) C) \( \frac{-\sin(x) x – 2\cos(x)}{x^3} \) D) \( \frac{\sin(x) x + 2\cos(x)}{x^3} \) **Answer:** A) \( \frac{-\sin(x) x^2 – 2x \cos(x)}{x^4} \) **Explanation:** Use the quotient rule: \( \frac{d}{dx}\left(\frac{\cos(x)}{x^2}\right) = \frac{(-\sin(x)) x^2 – \cos(x) \cdot 2x}{(x^2)^2} \). 50. **Question:** If \( y = x^2 e^x + e^x \), then \( \frac{dy}{dx} \) is: A) \( x^2 e^x + 2x e^x + e^x \) B) \( x^2 e^x + 2x e^x \) C) \( 2x e^x + e^x \) D) \( x^2 e^x + e^x \) **Answer:** A) \( x^2 e^x + 2x e^x + e^x \) **Explanation:** Differentiate term by term: \( \frac{d}{dx}(x^2 e^x) = 2x e^x + x^2 e^x \), \( \frac{d}{dx}(e^x) = e^x \). — ### **Notes:** – These questions cover key differentiation concepts such as the power rule, product rule, quotient rule, chain rule, and derivatives of trigonometric, exponential, and logarithmic functions. – They are designed to be objective, concise, and aligned with the TGT level, focusing on fundamental calculus skills. – If you need additional questions, specific subtopics (e.g., implicit differentiation, higher-order derivatives), or solutions in a different format, please let me know!