BCA 102 Module-II : Successive Differentiation

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Successive Differentiation (क्रमिक अवकलन)

🔍 Introduction (परिचय):

जब हम किसी function का derivative एक बार से अधिक निकालते हैं, यानी बार-बार differentiation करते हैं, तो इसे Successive Differentiation या Higher Order Differentiation कहा जाता है।

📌 इसे हम second derivative, third derivative, और आगे तक extend कर सकते हैं।

📚 Definition:

Let y=f(x) be a differentiable function.

First derivative:

​

Second derivative:

Third derivative:

In general:

​

🧠 Example 1:

1st derivative:

2nd derivative:

3rd derivative:

4th derivative:

📘 General Result for Polynomial:

🧠 Example 2:

Find first 3 derivatives of y = sin⁡x

1st derivative:

2nd derivative:

3rd derivative:

4th derivative:

📌 Thus: Trigonometric functions often follow a cyclic pattern in their higher-order derivatives.

⚙️ Useful Notation:

🎯 Objectives Recap:

  • Higher order derivative निकालना सीखना
  • Polynomial और trigonometric functions के लिए successive derivatives निकालना
  • Cyclic nature और pattern पहचानना

📝 Exercise (अभ्यास प्रश्न):

📌 Short Questions:

  1. Define second and third order derivatives.
  2. What is the 4th derivative of y=x^3 ?
  3. Show that the 4th derivative of y = sin⁡x is same as the function.

📌 Numerical Problems: