📐 Mathematics — Class XII · Calculus

Application of Derivatives

Where calculus meets the real world — rates, tangents, curves, and optimization

📖 Chapter 6 ⏱ ~75 min read 🏷 Calculus

In this chapter

Introduction
Rate of Change of Quantities
Increasing and Decreasing Functions
Tangents and Normals
Approximations
Maxima and Minima
Summary

6.1 Introduction

"With the Calculus as a key, Mathematics can be successfully applied to the explanation of the course of Nature." — Whitehead

In Chapter 5, we learned to differentiate composite, inverse trigonometric, implicit, exponential, and logarithmic functions. In this chapter, we apply these derivatives to solve real-world problems: determining rates of change, finding equations of tangents, locating maxima and minima, and making approximations.

6.2 Rate of Change of Quantities

Whenever one quantity y varies with another quantity x, satisfying y = f(x), the derivative dy/dx represents the rate of change of y with respect to x.

Figure 6.1 — Rate of change: how fast one quantity changes with respect to another

📐 Chain Rule for Rates

If x = f(t) and y = g(t), then:

dy/dx = (dy/dt) ÷ (dx/dt)

This lets us find the rate of y w.r.t. x even when both depend on a third variable like time.

6.3 Increasing and Decreasing Functions

The sign of the first derivative tells us whether a function is going up or down.

Figure 6.2 — f'(x) > 0 means increasing; f'(x) < 0 means decreasing

📈

Increasing

f'(x) ≥ 0 on (a, b). As x increases, f(x) increases. Graph goes uphill.

📉

Decreasing

f'(x) ≤ 0 on (a, b). As x increases, f(x) decreases. Graph goes downhill.

📐 Algorithm

Step 1: Find f'(x)
Step 2: Solve f'(x) = 0 to get critical points
Step 3: These points divide the domain into intervals
Step 4: Test the sign of f'(x) in each interval
Step 5: + means increasing, − means decreasing

6.4 Tangents and Normals

The derivative at a point gives the slope of the tangent to the curve at that point. The normal is perpendicular to the tangent.

Figure 6.3 — Tangent has slope f'(x₁); normal has slope −1/f'(x₁)

Equations Tangent: y - y₁ = f'(x₁)(x - x₁) Normal: y - y₁ = (-1/f'(x₁))(x - x₁)

⚠️ Special Cases

Horizontal tangent: f'(x₁) = 0 → tangent: y = y₁
Vertical tangent: f'(x₁) = ∞ → tangent: x = x₁

6.5 Approximations

Differentials allow us to approximate values of functions near a known point using the tangent line.

Figure 6.4 — Δy ≈ dy = f'(x) · Δx for small changes

📐 Linear Approximation

For small Δx:

f(x + Δx) ≈ f(x) + f'(x) · Δx

Example: √26 = √(25 + 1) ≈ 5 + (1/10)(1) = 5.1

6.6 Maxima and Minima

Finding the highest and lowest points on a curve — the crown jewel of optimization.

Figure 6.5 — First derivative test (sign change) and second derivative test (concavity)

⬆️

Local Maxima

f'(c) = 0 and f' changes + → − (or f''(c) < 0). The function peaks at c.

⬇️

Local Minima

f'(c) = 0 and f' changes − → + (or f''(c) > 0). The function valleys at c.

Critical Points & Working Rule

📐 Steps for Absolute Extrema on [a, b]

Step 1: Find all critical points where f'(x) = 0 or f'(x) doesn't exist
Step 2: Include the endpoints a and b
Step 3: Evaluate f at all these points
Step 4: The largest value is the absolute maximum; the smallest is the absolute minimum

6.7 Summary

✅ Key Takeaways

• dy/dx = rate of change of y w.r.t. x · chain rule: dy/dx = (dy/dt)/(dx/dt)

• f'(x) ≥ 0 → increasing · f'(x) ≤ 0 → decreasing

• Tangent: y − y₁ = f'(x₁)(x − x₁) · Normal: slope = −1/f'(x₁)

• Approximation: f(x + Δx) ≈ f(x) + f'(x)Δx for small Δx

• Critical point: where f'(x) = 0 or undefined

• First Derivative Test: sign change of f' at critical point → max/min

• Second Derivative Test: f'(c) = 0 and f''(c) < 0 → max; f''(c) > 0 → min

• Absolute extrema: compare f at critical points and endpoints

Ch 5 — Continuity and Differentiability Ch 7 — Integrals