Calculus - Differentiation and Integration

Subject: Additional Mathematics
Topic: 10
Cambridge Code: 4037 / 0606

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Introduction to Differentiation

Differentiation - Process of finding the rate of change of a function

Derivative - Rate of change of a function at a point

Notation

$\frac{dy}{dx} = f'(x) = \frac{df}{dx}$

All denote the derivative of $y$ with respect to $x$

Geometric Interpretation

The derivative is the slope of the tangent line to the curve at a point

Definition

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

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Rules of Differentiation

Rule 1: Power Rule

$\frac{d}{dx}(x^n) = nx^{n-1}$

Examples:

• $\frac{d}{dx}(x^3) = 3x^2$
• $\frac{d}{dx}(x^{-2}) = -2x^{-3}$
• $\frac{d}{dx}(x^{\frac{1}{2}}) = \frac{1}{2}x^{-\frac{1}{2}}$

Rule 2: Constant Multiple Rule

$\frac{d}{dx}(kf(x)) = k \frac{df}{dx}$

Example: $\frac{d}{dx}(5x^2) = 5 \cdot 2x = 10x$

Rule 3: Sum Rule

$\frac{d}{dx}(f(x) + g(x)) = \frac{df}{dx} + \frac{dg}{dx}$

Example: $\frac{d}{dx}(x^3 + 2x^2 - 5x + 1) = 3x^2 + 4x - 5$

Rule 4: Product Rule

$\frac{d}{dx}(u \cdot v) = u\frac{dv}{dx} + v\frac{du}{dx}$

Example: Differentiate $f(x) = x^2(3x+1)$

Let $u = x^2$, $v = 3x+1$

• $\frac{du}{dx} = 2x$, $\frac{dv}{dx} = 3$
• $\frac{df}{dx} = x^2(3) + (3x+1)(2x) = 3x^2 + 6x^2 + 2x = 9x^2 + 2x$

Rule 5: Quotient Rule

$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$

Example: Differentiate $f(x) = \frac{x^2}{x+1}$

Let $u = x^2$, $v = x+1$

• $\frac{du}{dx} = 2x$, $\frac{dv}{dx} = 1$
• $\frac{df}{dx} = \frac{(x+1)(2x) - x^2(1)}{(x+1)^2} = \frac{2x^2 + 2x - x^2}{(x+1)^2} = \frac{x^2 + 2x}{(x+1)^2}$

Rule 6: Chain Rule

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

Example: Differentiate $f(x) = (2x+1)^3$

Let $u = 2x+1$, then $f = u^3$

• $\frac{du}{dx} = 2$, $\frac{df}{du} = 3u^2$
• $\frac{df}{dx} = 3u^2 \cdot 2 = 6(2x+1)^2$

Special Derivatives

$\frac{d}{dx}(e^x) = e^x$

$\frac{d}{dx}(\ln x) = \frac{1}{x}$

$\frac{d}{dx}(\sin x) = \cos x$

$\frac{d}{dx}(\cos x) = -\sin x$

$\frac{d}{dx}(\tan x) = \sec^2 x$

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Applications of Differentiation

Finding Stationary Points

At stationary point: $\frac{dy}{dx} = 0$

Types:

• Maximum: $f''(x) < 0$
• Minimum: $f''(x) > 0$
• Inflection: $f''(x) = 0$ (and changes sign)

Example: Find Maximum of $f(x) = -x^2 + 4x$

$f'(x) = -2x + 4 = 0 \Rightarrow x = 2$

$f''(x) = -2 < 0 \Rightarrow \text{Maximum}$

$f(2) = -4 + 8 = 4$

Maximum point: $(2, 4)$

Rates of Change

Derivative represents instantaneous rate of change

Example: Distance $s = 2t^2 + 3t$ meters Velocity: $v = \frac{ds}{dt} = 4t + 3$ m/s

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Introduction to Integration

Integration - Reverse process of differentiation; finding the original function from its derivative

Indefinite Integral: $\int f(x) dx = F(x) + C$

where $F'(x) = f(x)$ and $C$ is the constant of integration

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Rules of Integration

Rule 1: Power Rule

$\int x^n dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$

Examples:

• $\int x^3 dx = \frac{x^4}{4} + C$
• $\int x^{-2} dx = \frac{x^{-1}}{-1} + C = -\frac{1}{x} + C$

Rule 2: Constant Multiple Rule

$\int kf(x) dx = k \int f(x) dx$

Example: $\int 5x^2 dx = 5 \int x^2 dx = 5 \cdot \frac{x^3}{3} + C = \frac{5x^3}{3} + C$

Rule 3: Sum Rule

$\int (f(x) + g(x)) dx = \int f(x) dx + \int g(x) dx$

Special Integrals

$\int e^x dx = e^x + C$

$\int \frac{1}{x} dx = \ln |x| + C$

$\int \sin x dx = -\cos x + C$

$\int \cos x dx = \sin x + C$

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Definite Integration

Definite Integral: Integral with limits of integration

$\int_a^b f(x) dx = [F(x)]_a^b = F(b) - F(a)$

Example

$\int_1^3 x^2 dx = \left[\frac{x^3}{3}\right]_1^3 = \frac{27}{3} - \frac{1}{3} = 9 - \frac{1}{3} = \frac{26}{3}$

Area Under Curve

Definite integral represents the area between curve and x-axis

$\text{Area} = \int_a^b f(x) dx$

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Key Points to Remember

1. Power rule most common differentiation rule
2. Chain rule for composite functions
3. Product/quotient rules for products/fractions
4. Integration is reverse of differentiation
5. Always add constant $C$ to indefinite integrals
6. Definite integral evaluates area

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Worked Examples

Example 1: Differentiation

Differentiate $f(x) = 3x^2 - 2x^{-1} + 5\sqrt{x}$

Rewrite: $f(x) = 3x^2 - 2x^{-1} + 5x^{\frac{1}{2}}$

$f'(x) = 6x + 2x^{-2} + \frac{5}{2}x^{-\frac{1}{2}} = 6x + \frac{2}{x^2} + \frac{5}{2\sqrt{x}}$

Example 2: Finding Maximum

Find maximum of $f(x) = x^3 - 3x^2 + 2$

$f'(x) = 3x^2 - 6x = 3x(x-2) = 0$ $x = 0 \text{ or } x = 2$

$f''(x) = 6x - 6$

• At $x = 0$: $f''(0) = -6 < 0$ (Maximum)
• At $x = 2$: $f''(2) = 6 > 0$ (Minimum)

Maximum: $(0, 2)$

Example 3: Integration

$\int (4x^3 - 2x + 5) dx = x^4 - x^2 + 5x + C$

Example 4: Definite Integration

$\int_0^2 (2x+1) dx = [x^2 + x]_0^2 = (4+2) - (0) = 6$

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Practice Questions

1. Differentiate:

   • $f(x) = x^4 - 3x^2 + 2$
   • $g(x) = (2x+3)^2$
   • $h(x) = \frac{x}{x^2+1}$

2. Find stationary points and determine their nature

3. Integrate:

   • $\int (x^3 + 2x - 1) dx$
   • $\int (2\sin x + e^x) dx$

4. Evaluate $\int_0^1 (3x^2 + 2x) dx$

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Revision Tips

• Power rule is fundamental: $\frac{d}{dx}(x^n) = nx^{n-1}$
• Chain rule for brackets/composite functions
• Integration is reverse of differentiation
• Don't forget constant $+C$ in indefinite integrals
• Definite integral gives area under curve