🔹 1. Introduction to Application of Derivatives

In earlier chapters, you learned how to find derivatives. In this chapter, you will learn how to use derivatives to solve practical problems.

Applications include:

• Finding slope of tangent and normal
• Determining increasing/decreasing behavior
• Finding maxima and minima
• Solving optimization problems
• Approximations and error estimation

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🔹 2. Rate of Change of Quantities

Derivative represents rate of change of one quantity with respect to another.

📌 Formula

$\frac{dy}{dx}$dxdy​

📖 Explanation

• If $y$y depends on $x$x, then $\frac{dy}{dx}$dxdy​ shows how fast $y$y changes as $x$x changes.

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📍 Example

If radius of a circle increases with time, then:

• Rate of change of area:

$$\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$$dtdA​=2πrdtdr​

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🔹 3. Increasing and Decreasing Functions

Derivative helps determine whether a function is increasing or decreasing.

📌 Conditions

$\text{If } \frac{dy}{dx} > 0 \Rightarrow \text{Increasing function}$If dxdy​>0⇒Increasing function

$\text{If } \frac{dy}{dx} < 0 \Rightarrow \text{Decreasing function}$If dxdy​<0⇒Decreasing function

📖 Interpretation

• Positive derivative → function rises
• Negative derivative → function falls

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📍 Example

For $f(x) = x^2$f(x)=x2:

• $f'(x) = 2x$f′(x)=2x
• Increasing when $x > 0$x>0
• Decreasing when $x < 0$x<0

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🔹 4. Tangents and Normals

✅ Tangent

A tangent touches the curve at a point.

📌 Equation of Tangent

$y – y_1 = m(x – x_1)$y−y1​=m(x−x1​)-10-8-6-4-2246810-10-5510-8.00, -8.008.00, 8.00m = 1.00

Where:

• $m = \frac{dy}{dx}$m=dxdy​ at the point

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✅ Normal

A normal is perpendicular to the tangent.

📌 Slope of Normal

$$m_{normal} = -\frac{1}{m_{tangent}}$$mnormal​=−mtangent​1​

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📍 Example

Find tangent to $y = x^2$y=x2 at $x = 1$x=1

• $dy/dx = 2x \Rightarrow m = 2$dy/dx=2x⇒m=2

Equation:$$y – 1 = 2(x – 1)$$y−1=2(x−1)

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🔹 5. Approximation and Errors

Derivatives help in approximation.

📌 Formula

$$dy = f'(x) dx$$dy=f′(x)dx

📖 Interpretation

• Small change in $y$y can be estimated using derivative

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📍 Example

Find approximate change in $y = x^2$y=x2 when $x$x changes from 2 to 2.01

• $dy = 2x dx = 2(2)(0.01) = 0.04$dy=2xdx=2(2)(0.01)=0.04

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🔹 6. Maxima and Minima

This is one of the most important topics in AOD.

🔹 Critical Points

Points where:$$f'(x) = 0 \text{ or undefined}$$f′(x)=0 or undefined

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🔹 First Derivative Test

• If derivative changes from + to − → Maximum
• If derivative changes from − to + → Minimum

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🔹 Second Derivative Test

📌 Formula

$f”(x) > 0 \Rightarrow \text{Minimum}, \quad f”(x) < 0 \Rightarrow \text{Maximum}$f′′(x)>0⇒Minimum,f′′(x)<0⇒Maximum

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📍 Example

Find maxima/minima of $f(x) = x^2$f(x)=x2

• $f'(x) = 2x = 0 \Rightarrow x = 0$f′(x)=2x=0⇒x=0
• $f”(x) = 2 > 0$f′′(x)=2>0

👉 Minimum at $x = 0$x=0

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🔹 7. Optimization Problems

Optimization means finding maximum or minimum values in real-life situations.

📌 Steps to Solve

1. Identify variables
2. Form equation
3. Differentiate
4. Set derivative = 0
5. Verify using second derivative

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📍 Example

Find dimensions of rectangle with maximum area for fixed perimeter.

(Solved using derivative and optimization steps.)

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🔹 8. Monotonicity

Monotonic functions are always increasing or decreasing.

• Strictly increasing → derivative always positive
• Strictly decreasing → derivative always negative

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🔹 9. Important Formulas Summary

📌 Key Results

• $\frac{dy}{dx}$dxdy​ → rate of change
• Increasing → $f'(x) > 0$f′(x)>0
• Decreasing → $f'(x) < 0$f′(x)<0
• Tangent slope → $dy/dx$dy/dx
• Normal slope → $-1/(dy/dx)$−1/(dy/dx)
• Maxima/Minima → $f'(x)=0$f′(x)=0

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🔹 10. Graphical Interpretation

• Positive slope → graph rising
• Negative slope → graph falling
• Zero slope → horizontal tangent
• Turning point → maxima/minima

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🔹 11. Real-Life Applications

Application of derivatives is used in:

• Physics (velocity, acceleration)
• Economics (profit maximization)
• Engineering (design optimization)
• Business (cost minimization)

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🔹 12. Important Questions for Practice

1. Find intervals where function is increasing/decreasing
2. Find tangent and normal equations
3. Solve maxima-minima problems
4. Solve optimization word problems
5. Approximation-based questions

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🔹 13. Common Mistakes

• Forgetting chain rule
• Wrong sign in second derivative test
• Not checking endpoints in optimization
• Confusing maxima and minima

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🔹 14. Exam Tips

• Practice graphs
• Focus on maxima-minima
• Revise formulas daily
• Solve NCERT thoroughly
• Show proper steps

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🔹 15. Conclusion

Application of Derivatives is a high-weightage and concept-based chapter. Once you understand the logic behind derivatives, this chapter becomes easy and scoring.

Master this chapter by:

• Practicing problems
• Understanding graphs
• Applying concepts to real-life situations