CBSE Class 12 Mathematics: Calculus — Differentiation — Important Questions with Answers 2026

Short Answer Questions (2-3 Marks)

• Q: Differentiate: y = sin(x²). State the chain rule.
  Ans: Chain Rule: If y = f(g(x)), then dy/dx = f'(g(x)) × g'(x). For y = sin(x²): dy/dx = cos(x²) × 2x = 2x cos(x²). The chain rule is used when we have a composite function — differentiate the outer function, then multiply by the derivative of the inner function.
• Q: Find dy/dx if x² + y² = 25 (implicit differentiation).
  Ans: Differentiate both sides with respect to x: 2x + 2y(dy/dx) = 0. 2y(dy/dx) = -2x. dy/dx = -x/y. At point (3, 4): dy/dx = -3/4. This is the slope of the tangent to the circle at that point.
• Q: Find the maximum and minimum values of f(x) = x³ - 3x² + 4.
  Ans: f'(x) = 3x² - 6x = 3x(x-2). Setting f'(x) = 0: x = 0 or x = 2. f''(x) = 6x - 6. At x = 0: f''(0) = -6 < 0 → local maximum. f(0) = 4. At x = 2: f''(2) = 6 > 0 → local minimum. f(2) = 8 - 12 + 4 = 0. Maximum value = 4 at x = 0. Minimum value = 0 at x = 2.

Long Answer / Application Questions (4-6 Marks)

• Q: Find the equation of the tangent to y = x² - 4x + 5 at x = 2.
  Ans: At x = 2: y = 4 - 8 + 5 = 1. Point: (2, 1). dy/dx = 2x - 4. At x = 2: slope m = 0. Equation: y - 1 = 0(x - 2) → y = 1 (horizontal tangent).

Exam Tips for This Chapter

• Revise all definitions and laws from Calculus — Differentiation — commonly asked as 1-2 mark questions
• Practice diagrams related to Calculus — Differentiation — neat labelled diagrams carry 2-3 marks
• For numericals, always show formula → substitution → answer with correct units
• Previous year analysis shows Calculus — Differentiation carries significant marks in the board exam