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5.Derivatives from First Principles
Class 11 • Mathematics • NCERT
Derivatives from First Principles
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This lesson explains how to find derivatives using first principles as prescribed by NCERT Class 11. Students learn the definition clearly and apply it step by step to simple functions.
Lesson Objectives
- Understand the definition of derivative from first principles.
- Apply the definition to basic functions.
- Find derivatives of constants and simple polynomials.
- Build a strong base for derivative rules.
1. Meaning of First Principles
First principles means finding the derivative directly from its definition using limits, without using shortcut formulas.
This method shows where derivative formulas come from.
2. Definition of Derivative (First Principles)
The derivative of a function f(x) at x is defined as:
f′(x) = limh→0 [f(x + h) − f(x)] / h
3. Derivative of a Constant Function
Let f(x) = c, where c is a constant.
f′(x) = limh→0 (c − c)/h = 0
So, the derivative of a constant is zero.
4. Derivative of f(x) = x
Let f(x) = x.
f′(x) = limh→0 [(x + h) − x]/h = 1
5. Derivative of f(x) = x²
Let f(x) = x².
f′(x) = limh→0 [(x + h)² − x²]/h
= limh→0 (2x + h) = 2x
= limh→0 (2x + h) = 2x
6. Important NCERT Observations
• First principles use limits
• Method is lengthy but fundamental
• Used to derive standard derivative formulas
• Method is lengthy but fundamental
• Used to derive standard derivative formulas
Practice Questions (NCERT)
- What is meant by derivative from first principles?
- Write the definition of derivative using limits.
- Find the derivative of f(x) = 5 using first principles.
- Find the derivative of f(x) = x using first principles.
- Find the derivative of f(x) = x² using first principles.
- What is the derivative of a constant?
- Why is h taken to zero?
- Is this method used for all functions?
- Why are first principles important?
- Is this topic part of NCERT Class 11 syllabus?
✅ Show Answer Key
- Finding derivative using definition and limits.
- limh→0 [f(x + h) − f(x)] / h
- 0
- 1
- 2x
- Zero
- To find instantaneous rate of change.
- No
- They form the basis of derivative formulas.
- Yes
© Aviate Learning – Limits & Derivatives (NCERT Class 11)