Second Order Derivatives | 5 Marks Most Important Questions | II PUC Maths Karnataka Board 2025–26

📘 5 Marks Important Questions – Second Order Derivatives | One Shot Revision
II PUC Mathematics | Karnataka State Board | Board Exam Focus

In this video, we solve the Most Important 5 Marks Questions from the topic
Second Order Derivatives for II PUC Mathematics – Karnataka State Board.

This one-shot session covers frequently asked “Prove That” type problems and
second derivative applications that are very important for board examinations.

🔥 Topics Covered

• Second Order Derivatives of exponential functions
• Second Order Derivatives of inverse trigonometric functions
• Logarithmic differentiation problems
• Implicit differentiation
• Prove-type questions involving d²y/dx²
• Standard board-exam pattern questions

🔥 Problems Covered with Timeline

1) If y = Ae^(mx) + Be^(nx), show that
d²y/dx² − (m + n)(dy/dx) + mn y = 0.

2) If y = 3e^(2x) + 2e^(3x), prove that
d²y/dx² − 5(dy/dx) + 6y = 0.
⏱ 6:52

3) If y = 3cos(log x) + 4sin(log x), prove that
x²(d²y/dx²) + x(dy/dx) + y = 0.
⏱ 11:20

4) If y = 5cos(log x) + 7sin(log x), prove that
x²(d²y/dx²) + x(dy/dx) + y = 0.
⏱ 18:51

5) If y = tan^(-1)x, prove that
(1 + x²)(d²y/dx²) + 2x(dy/dx) = 0.
⏱ 23:09

6) If y = (tan^(-1)x)², prove that
(x² + 1)²(d²y/dx²) + 2x(x² + 1)(dy/dx) = 2.
⏱ 25:55

7) If y = e^(tan^(-1)x), prove that
(1 + x²)²(d²y/dx²) + 2x(dy/dx) − y = 0.
⏱ 30:35

8) If y = e^(tan^(-1)x), prove that
(1 + x²)(d²y/dx²) + (2x − 1)(dy/dx) = 0.
⏱ 34:14

9) If y = sin^(-1)x, prove that
(1 − x²)(d²y/dx²) − x(dy/dx) = 0.
⏱ 37:56

10) If y = (sin^(-1)x)², prove that
(1 − x²)(d²y/dx²) − x(dy/dx) = 2.
⏱ 41:53

11) If y = e^[a(cos^(-1)x)], prove that
(1 − x²)(d²y/dx²) − x(dy/dx) = a²y.
⏱ 44:57

12) If (e^y)(x + 1) = 1, prove that
dy/dx = e^y
and hence prove that
d²y/dx² = (dy/dx)²
⏱ 50:24

📌 What You Will Learn

✔ How to compute second derivatives step by step
✔ How to simplify expressions before differentiating again
✔ Board-exam answer writing method
✔ Common mistakes in second derivative problems
✔ How to score full 5 marks confidently

🎯 Who Should Watch

• II PUC students – Karnataka State Board
• Students preparing for board examinations
• Students revising Applications of Derivatives
• Students aiming for full marks in 5-mark questions

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