1. Introduction to Application of Derivatives
The derivative of a function is not only used to calculate the slope of a curve. It also helps us understand how a function behaves. This is the main idea behind Application of Derivatives.
If y = f(x) then f'(x) = dy/dx represents the instantaneous rate of change of y with respect to x.
Measures how rapidly one quantity changes with another.
The sign of the derivative tells whether a function rises or falls.
Derivatives help locate important high and low points.
2. Rate of Change
Instantaneous Rate of Change
If y = f(x) then the rate of change of y with respect to x is
For example, if the area of a circle is
then
Graphical Meaning of Rate of Change
Related Rates
When two or more quantities depend on time, differentiate the relation with respect to time.
2x dx/dt + 2y dy/dt = 0
3. Increasing and Decreasing Functions
Increasing Function
A function f is increasing on an interval I if
In simple words: as x increases, the value of f(x) increases.
Decreasing Function
A function f is decreasing on an interval I if
In simple words: as x increases, the value of f(x) decreases.
Graph: Increasing Function
Graph: Decreasing Function
4. Strictly Increasing and Strictly Decreasing
Strictly Increasing
A function is strictly increasing on an interval if for every pair x₁ < x₂ we have
The function never decreases as x moves from left to right. Distinct x-values give distinct function values in the direction of increase.
Strictly Decreasing
A function is strictly decreasing on an interval if
As we move from left to right, the function value always decreases.
Difference to Remember
| Concept | Meaning | Derivative indication |
|---|---|---|
| Increasing | Function values rise as x increases. | f'(x) ≥ 0 |
| Strictly Increasing | Every larger x gives a strictly larger value. | f'(x) > 0 |
| Decreasing | Function values fall as x increases. | f'(x) ≤ 0 |
| Strictly Decreasing | Every larger x gives a strictly smaller value. | f'(x) < 0 |
5. Tangent and Normal
Tangent
The tangent to a curve at a point is the straight line that touches the curve at that point and has the same instantaneous slope as the curve.
Slope of tangent = f'(x₁)
Therefore,
y-y₁ = f'(x₁)(x-x₁)
Normal
The normal is the line perpendicular to the tangent at the point of contact.
Slope of normal = -1 / f'(x₁)
Hence,
y-y₁ = -1/f'(x₁)(x-x₁)
Graphical Representation of Tangent and Normal
Slope = f'(x₁)
Slope = −1/f'(x₁)
Tangent and normal are perpendicular.
6. Critical and Stationary Points
A critical point of a function occurs at a point in its domain where the derivative is zero or does not exist.
Stationary Point
A point where f'(c)=0 is called a stationary point.
Important Warning
Every maximum or minimum of a differentiable function at an interior point is stationary, but every stationary point need not be a maximum or minimum.
Example: Stationary Point That Is Not an Extremum
Consider f(x)=x³. At x=0, f'(0)=0, but the function continues increasing through the point.
7. Local Maxima and Local Minima
Local Maximum
A function has a local maximum at x=c if
The function reaches a high point compared with nearby points.
Local Minimum
A function has a local minimum at x=c if
The function reaches a low point compared with nearby points.
Graphical Explanation
8. Absolute Maxima and Absolute Minima
Absolute Maximum
A function has an absolute maximum at x=a if
It is the greatest function value over the entire domain.
Absolute Minimum
A function has an absolute minimum at x=b if
It is the smallest function value over the entire domain.
Local vs Absolute: Graphical Understanding
9. First Derivative Test
The first derivative test determines whether a critical point is a local maximum or local minimum by observing the sign of f'(x) on both sides.
Increasing → Decreasing
Decreasing → Increasing
No change of sign
First Derivative Sign Chart
| Interval | (-∞, c) | c | (c, ∞) |
|---|---|---|---|
| f'(x) | + | 0 | − |
| Behaviour | Increasing | Maximum | Decreasing |
10. Second Derivative Test
Local Maximum
Concave downward at the stationary point.
Local Minimum
Concave upward at the stationary point.
Inconclusive
Use the first derivative test or another suitable method.
Concave Up → Possible Minimum
Concave Down → Possible Maximum
11. Optimization Using Derivatives
Optimization problems ask us to find the largest or smallest possible value of a quantity.
Standard Procedure
- Identify the quantity to maximize or minimize.
- Express it in one variable.
- Find the domain or restrictions.
- Differentiate.
- Put the first derivative equal to zero.
- Find critical points.
- Use first or second derivative test.
- Check endpoints when required.
- State the final answer with units.
Fixed Perimeter Rectangle
Important Optimization Result
If a rectangle has fixed perimeter P then its area is maximum when the rectangle becomes a square.x = y = P/4
12. Solved Examples
Find the rate of change of area of a circle with respect to its radius.
Given
Differentiating with respect to r:
The radius of a circle increases at 2 cm/s. Find the rate at which its area increases when r = 5 cm.
dA/dt = 2πr dr/dt
= 2π(5)(2)
= 20π
Find the tangent to y = x² + 2x + 1 at x = 1.
dy/dx = 2x + 2
At x = 1:
m = 4
y = 1 + 2 + 1 = 4
Point = (1,4)
Tangent:
y - 4 = 4(x - 1)
y = 4x
Find the normal to y = x² + 2x + 1 at x = 1.
Normal slope: mₙ = -1/4
Point = (1,4)
Therefore,
y - 4 = -1/4(x - 1)
Determine the intervals where f(x) = x³ − 3x is increasing and decreasing.
= 3(x - 1)(x + 1)
Critical numbers: x = -1, 1
| Interval | f'(x) | Behaviour |
|---|---|---|
| (-∞,-1) | + | Increasing |
| (-1,1) | − | Decreasing |
| (1,∞) | + | Increasing |
Show that x = 0 is not a maximum or minimum for f(x)=x³.
f'(0)=0
However, 3x²>0 on both sides of zero except at zero. Therefore, the function remains increasing.
Find the local maxima and minima of f(x)=x³−3x²+2.
=3x(x−2)
Critical points: x=0, 2
f''(x)=6x−6
f''(0)=-6<0
Therefore local maximum.
f(0)=2
f''(2)=6>0
Therefore local minimum.
f(2)=-2
Find the absolute maximum and minimum of f(x)=x³−3x on [-2,2].
Critical points: x = -1, 1
| x | -2 | -1 | 1 | 2 |
|---|---|---|---|---|
| f(x) | -2 | 2 | -2 | 2 |
A rectangle has perimeter 40 cm. Find its maximum possible area.
2x+2y=40
y=20-x
Area:
A=x(20-x)
A=20x-x²
dA/dx=20-2x
20-2x=0
x=10
y=10
d²A/dx²=-2<0
13. Practice Questions
Q1. Find the derivative representing the rate of change of the area A = πr² with respect to r.
Try this question first. Then verify your solution using the relevant derivative test or formula.
Q2. If r increases at 3 cm/s, find dA/dt when r = 4 cm.
Try this question first. Then verify your solution using the relevant derivative test or formula.
Q3. Find the tangent to y = x² at x = 2.
Try this question first. Then verify your solution using the relevant derivative test or formula.
Q4. Find the normal to y = x² at x = 2.
Try this question first. Then verify your solution using the relevant derivative test or formula.
Q5. Find the point where the tangent to y = x² is parallel to the line y = 4x + 1.
Try this question first. Then verify your solution using the relevant derivative test or formula.
Q6. Determine where f(x)=x³−3x is increasing and decreasing.
Try this question first. Then verify your solution using the relevant derivative test or formula.
Q7. Find the critical points of f(x)=x³−6x²+9x.
Try this question first. Then verify your solution using the relevant derivative test or formula.
Q8. Show that x=0 is not an extremum of f(x)=x³.
Try this question first. Then verify your solution using the relevant derivative test or formula.
Q9. Find local maxima and minima of f(x)=x³−3x²+2.
Try this question first. Then verify your solution using the relevant derivative test or formula.
Q10. Use the second derivative test for f(x)=x²−6x+5.
Try this question first. Then verify your solution using the relevant derivative test or formula.
Q11. Find the absolute maximum and minimum of f(x)=x²−4x+1 on [0,5].
Try this question first. Then verify your solution using the relevant derivative test or formula.
Q12. A rectangle has perimeter 60 cm. Find the dimensions for maximum area.
Try this question first. Then verify your solution using the relevant derivative test or formula.
Q13. Two positive numbers have sum 20. Find the numbers for which their product is maximum.
Try this question first. Then verify your solution using the relevant derivative test or formula.
Q14. Find the intervals of increase and decrease for f(x)=x⁴−4x².
Try this question first. Then verify your solution using the relevant derivative test or formula.
Q15. Find the tangent to y=x³ at x=1.
Try this question first. Then verify your solution using the relevant derivative test or formula.
Q16. Find the normal to y=x²+1 at x=1.
Try this question first. Then verify your solution using the relevant derivative test or formula.
Q17. Explain the difference between local maximum and absolute maximum.
Try this question first. Then verify your solution using the relevant derivative test or formula.
Q18. Explain the difference between increasing and strictly increasing functions.
Try this question first. Then verify your solution using the relevant derivative test or formula.
Q19. Find the maximum value of x(10−x) for 0≤x≤10.
Try this question first. Then verify your solution using the relevant derivative test or formula.
Q20. Find the minimum distance of the point (2,0) from the line y=x.
Try this question first. Then verify your solution using the relevant derivative test or formula.
14. HOT Questions
A stationary point need not be a point of maximum or minimum. Explain this statement using an appropriate example and graph.
Prove using derivatives that among all rectangles having a fixed perimeter, the square has maximum area.
A function is increasing on an interval but its derivative may be zero at some isolated points. Explain why this does not necessarily contradict the idea of increase.
Explain why the endpoints of a closed interval must be checked when finding absolute maximum and minimum values.
Explain geometrically why the slopes of tangent and normal are negative reciprocals when both slopes are finite.
15. Quick Revision
| Concept | Key Formula / Condition |
|---|---|
| Rate of Change | dy/dx = f'(x) |
| Increasing | f'(x) ≥ 0 under suitable conditions |
| Strictly Increasing | x₁<x₂ ⇒ f(x₁)<f(x₂) |
| Decreasing | f'(x) ≤ 0 under suitable conditions |
| Strictly Decreasing | x₁<x₂ ⇒ f(x₁)>f(x₂) |
| Tangent Slope | f'(x₁) |
| Tangent | y−y₁=f'(x₁)(x−x₁) |
| Normal Slope | −1/f'(x₁) |
| Local Maximum | f' changes + to − |
| Local Minimum | f' changes − to + |
| Second Derivative Maximum | f'(c)=0, f''(c)<0 |
| Second Derivative Minimum | f'(c)=0, f''(c)>0 |
| Absolute Maximum | Greatest value on entire domain/interval |
| Absolute Minimum | Smallest value on entire domain/interval |
Application of Derivatives — Visual Summary
f'(x) > 0
Function increases.
f'(x) < 0
Function decreases.
f''(c) < 0
Stationary local maximum.
f''(c) > 0
Stationary local minimum.
16. Exam Tips
Do This
- Differentiate carefully.
- Find all critical points.
- Check the domain before interpreting critical points.
- Use a sign chart for the first derivative test.
- Check endpoints for absolute extrema on closed intervals.
- Write tangent and normal equations step by step.
- Include units in rate-of-change and optimization answers.
- Clearly state whether an extremum is local or absolute.
Common Mistakes
- Do not assume every point where f'(x)=0 is a maximum or minimum.
- Do not forget points where f'(x) is undefined but the function is defined.
- Do not confuse local maximum with absolute maximum.
- Do not forget the negative sign in the normal slope.
- Do not ignore endpoint values in absolute extrema problems.
- Do not omit units from physical rate problems.
AOD Problem-Solving Flow
Master Application of Derivatives
Understand the graph, derivative sign, tangent, normal, local extrema and absolute extrema — and solve CBSE-style application problems with confidence.