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4.Introduction to Derivatives
Class 11 • Mathematics • NCERT
Introduction to Derivatives
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This lesson introduces the concept of derivatives as per NCERT Class 11. Students understand derivatives as rate of change and as the slope of a tangent, building a strong foundation for differential calculus.
Lesson Objectives
- Understand the meaning of a derivative.
- Interpret derivative as rate of change.
- Interpret derivative as slope of a tangent.
- Link derivatives with limits.
1. What Is a Derivative?
The derivative of a function measures how the function changes when its input changes.
Derivative tells us how fast or how slow a quantity is changing.
2. Derivative as Rate of Change
In real life, derivatives represent rates such as speed, acceleration, and growth.
Speed = Rate of change of distance with respect to time
3. Derivative as Slope of Tangent
Geometrically, the derivative of a function at a point represents the slope of the tangent to the curve at that point.
Slope of tangent = Instantaneous rate of change
4. Derivative Using Limits
The derivative of a function f(x) at x is defined using limits.
f′(x) = limh→0 [f(x + h) − f(x)] / h
5. Notations of Derivative
Different notations are used to represent derivatives.
• f′(x)
• dy/dx
• D[f(x)]
• dy/dx
• D[f(x)]
6. Simple NCERT Examples
• If distance = x², then speed = derivative of x²
• Slope of curve y = x² at x = 1 is positive
Practice Questions (NCERT)
- What is a derivative?
- What does a derivative represent physically?
- What does a derivative represent geometrically?
- Write the definition of derivative using limits.
- What is meant by instantaneous rate of change?
- Name any two notations of derivative.
- What is the derivative of a constant?
- Does derivative always exist?
- Why are derivatives important?
- Is this topic part of NCERT Class 11 syllabus?
✅ Show Answer Key
- Measure of change of a function.
- Rate of change.
- Slope of tangent.
- limh→0 [f(x + h) − f(x)] / h
- Rate of change at a specific point.
- f′(x), dy/dx
- Zero
- No
- To study change in quantities.
- Yes
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