Differentiation and Integration (Calculus)

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your WAEC exam.

Differentiation - The Derivative:

The derivative of $y$ with respect to $x$ is the rate of change of $y$ with respect to $x$.

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

Standard Derivatives:

Function	Derivative
$x^n$	$nx^{n-1}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$e^x$	$e^x$
$\ln x$	$\frac{1}{x}$
$e^{kx}$	$ke^{kx}$

Rules:

1. Sum Rule: $\frac{d}{dx}(u + v) = \frac{du}{dx} + \frac{dv}{dx}$

2. Product Rule: $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$

3. Quotient Rule: $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$

4. Chain Rule: $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$

⚡ WAEC Tip: For polynomials, just apply the power rule term by term. For $\frac{d}{dx}(5x^3 + 2x) = 15x^2 + 2$. It’s that simple.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Differentiation Examples:

Example 1: Find $\frac{dy}{dx}$ if $y = 3x^4 + 2x - 5$

$$\frac{dy}{dx} = 12x^3 + 2$$

Example 2: Find derivative of $y = (2x + 1)^3$

Using chain rule: $\frac{dy}{dx} = 3(2x+1)^2 \times 2 = 6(2x+1)^2$

Example 3: Find derivative of $y = x^2 \sin x$

Using product rule: $u = x^2$, $\frac{du}{dx} = 2x$ $v = \sin x$, $\frac{dv}{dx} = \cos x$ $$\frac{dy}{dx} = x^2 \cos x + 2x \sin x$$

Example 4: Find derivative of $y = \frac{x^2}{x+1}$

Using quotient rule with $u=x^2$, $v=x+1$: $$\frac{dy}{dx} = \frac{(x+1)(2x) - x^2(1)}{(x+1)^2} = \frac{2x^2 + 2x - x^2}{(x+1)^2} = \frac{x^2 + 2x}{(x+1)^2}$$

Applications of Differentiation:

1. Gradient of a Curve: The derivative at a point gives the gradient of the tangent.

2. Equations of Tangent and Normal:

• Tangent at $(x_1, y_1)$: $y - y_1 = m(x - x_1)$ where $m = f’(x_1)$
• Normal is perpendicular: $y - y_1 = -\frac{1}{m}(x - x_1)$

3. Turning Points (Maxima and Minima): At turning points, $\frac{dy}{dx} = 0$.

• If $\frac{d^2y}{dx^2} > 0$: Minimum point
• If $\frac{d^2y}{dx^2} < 0$: Maximum point

Worked Example: Find turning points of $y = x^3 - 3x^2 - 9x + 5$

$\frac{dy}{dx} = 3x^2 - 6x - 9 = 0$ $3(x^2 - 2x - 3) = 0$ $3(x-3)(x+1) = 0$ $x = 3$ or $x = -1$

$\frac{d^2y}{dx^2} = 6x - 6$

At $x = 3$: $\frac{d^2y}{dx^2} = 12 > 0$ → Minimum At $x = -1$: $\frac{d^2y}{dx^2} = -12 < 0$ → Maximum

⚡ Common Mistake: Forgetting the chain rule when differentiating functions of functions. $d/dx[(3x+1)^4] \neq 4(3x+1)^3$; it IS $4(3x+1)^3 \times 3 = 12(3x+1)^3$.

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory for serious exam preparation.

Integration - The Antiderivative:

Integration is the reverse of differentiation.

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

Standard Integrals:

Function	Integral
$x^n$	$\frac{x^{n+1}}{n+1} + C$
$\sin x$	$-\cos x + C$
$\cos x$	$\sin x + C$
$e^x$	$e^x + C$
$\frac{1}{x}$	$\ln
$e^{kx}$	$\frac{1}{k}e^{kx} + C$

Rules:

1. Sum Rule: $\int (u + v), dx = \int u, dx + \int v, dx$

2. Constant Multiple: $\int kf(x), dx = k \int f(x), dx$

Integration Examples:

Example 1: $\int (3x^2 + 2x - 4), dx$

$$= \frac{3x^3}{3} + \frac{2x^2}{2} - 4x + C = x^3 + x^2 - 4x + C$$

Example 2: $\int \frac{1}{x}, dx = \ln|x| + C$

Example 3: $\int \sin(2x), dx$

Let $u = 2x$, then $du = 2dx$, so $dx = \frac{du}{2}$ $$\int \sin(2x), dx = \int \sin u \cdot \frac{du}{2} = -\frac{1}{2}\cos u + C = -\frac{1}{2}\cos(2x) + C$$

Definite Integrals:

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

Worked Example: $$\int_0^2 (x^2 + 1), dx = \left[\frac{x^3}{3} + x\right]_0^2 = \left(\frac{8}{3} + 2\right) - 0 = \frac{14}{3}$$

Area Under a Curve:

The definite integral from $a$ to $b$ gives the area under $y = f(x)$ above the x-axis.

For area between curve and x-axis where $f(x) < 0$, the integral gives a negative value. Take absolute value of negative areas.

Area Between Two Curves: Area = $\int_a^b [f(x) - g(x)], dx$ (where $f(x) > g(x)$)

Kinematics:

If $v = \frac{ds}{dt}$ and $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$

• $s$ = displacement, $v$ = velocity, $a$ = acceleration, $t$ = time

$$\int a, dt = v + C$$ $$\int v, dt = s + C$$

Differential Equations:

$\frac{dy}{dx} = f(x)$ means $y = \int f(x), dx$

Worked Example: Given $\frac{dy}{dx} = 3x^2 + 2$ and $y = 5$ when $x = 1$, find $y$.

$y = \int (3x^2 + 2), dx = x^3 + 2x + C$

Using $x = 1, y = 5$: $5 = 1 + 2 + C$, so $C = 2$ $y = x^3 + 2x + 2$

Second Derivatives: $\frac{d^2y}{dx^2}$ tells us about the concavity of a curve.

• If $\frac{d^2y}{dx^2} > 0$: curve is concave up (U-shaped)
• If $\frac{d^2y}{dx^2} < 0$: curve is concave down (∩-shaped)

Points where $\frac{d^2y}{dx^2} = 0$ and concavity changes = Point of inflection.

⚡ WAEC Previous Year Pattern:

Year	Question	Concept
2023	Differentiate polynomial	Power rule
2022	Find turning points	dy/dx = 0
2021	Evaluate definite integral	F(b) - F(a)

Integration as Summation: The definite integral $\int_a^b f(x), dx$ represents the limit of a sum of areas of rectangles under a curve as width → 0.

This is the fundamental theorem connecting differentiation and integration.

Volumes of Revolution:

Volume generated when area under $y = f(x)$ from $x = a$ to $x = b$ is rotated 360° about x-axis: $$V = \pi \int_a^b y^2, dx$$

⚡ Exam Strategy: Always include $+C$ (constant of integration) when doing indefinite integration. Without $C$, your answer is incomplete. For rate problems, identify what rate is given, integrate to find the quantity, use given conditions to find the constant.

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