Integration

Think of a derivative as the rate at which something changes. If you're driving from Lagos to Ibadan, your derivative tells you your speed at any exact moment—not your average speed for the whole journey, but right now. In mathematics, the derivative of a function shows how fast the function's output changes when its input changes slightly.

When we write f'(x) or df/dx, we're asking: "How much does this function change for a tiny change in x?" For example, if a trader's profit function is P(x) = 5x², the derivative P'(x) = 10x tells us how quickly profit increases as more items are sold at any production level x.

Calculating derivatives uses specific rules: the power rule, product rule, and chain rule. These are your tools for finding derivatives quickly during exams.

When you differentiate a polynomial, you're finding the rate at which it changes. Think of it like measuring how fast a car is moving at any exact moment. For a polynomial term like x³, bring down the power as a coefficient and reduce the power by one, giving 3x². The same applies to each term separately.

For example, if a trader's profit function is P(x) = 5x² + 3x + 10 (where x represents items sold), differentiating gives P'(x) = 10x + 3. This tells the trader how much profit increases for each additional item sold.

Constants disappear when you differentiate because they don't change—they have zero rate of change. Always apply the power rule systematically to each term, and you'll master this concept completely.

When you differentiate sine, cosine, and tangent functions, you follow specific rules that you must memorize for WAEC. The derivative of sin(x) is cos(x), while the derivative of cos(x) is negative sin(x), written as –sin(x). For tan(x), the derivative is sec²(x). These rules work because trigonometric functions describe oscillating patterns, like the regular rise and fall of ocean tides along Nigeria's coastline or the swinging motion of a pendulum in a physics lab.

When functions combine with trigonometric expressions, such as sin(3x) or cos(2x + 5), you must apply the chain rule alongside these basic derivatives. This means multiplying by the derivative of the inner function. Understanding these patterns helps you solve real motion and wave problems.

When you see equations like 3x² + 2y² = 12, you cannot easily solve for y first. This is an implicit function, meaning y is hidden inside the equation. The clever trick is to differentiate both sides with respect to x, treating y as a function of x.

For example, if a circular plot of land in Lagos follows the equation x² + y² = 100, and you need the rate of change at a point, differentiate both sides. You get 2x + 2y(dy/dx) = 0. Notice how dy/dx appears when you differentiate y² using the chain rule—this is the product of the derivative of the outer function and dy/dx.

Rearranging gives dy/dx = -x/y. This tells you the slope at any point on that circle without solving y explicitly.

Transcendental functions are special mathematical functions that include exponential, logarithmic, and trigonometric expressions. When you differentiate these functions, you're finding their rate of change, which is crucial for solving real-world problems.

For example, if a Lagos bank calculates compound interest using the exponential function A = Pe^(rt), differentiating this helps determine how quickly your money grows at any moment. Similarly, when you differentiate sin(x), you get cos(x), and differentiating e^x gives you e^x again—a unique property that makes exponential functions powerful in mathematics.

The key formulas you must memorize are: the derivative of e^x is e^x, the derivative of ln(x) is 1/x, and the derivatives of sine and cosine functions involve switching between them with sign changes.

When you take the derivative of a function, you get the rate of change. Taking the derivative again gives you the second derivative, which tells you how quickly that rate is changing. This is crucial for finding maximum and minimum points.

Think of a trader in Lagos selling phone credit. If profit increases then decreases, there's a sweet spot where profit is highest—that's a maximum. The second derivative test helps you confirm whether a critical point is a peak or a valley. When the second derivative is negative, you have a maximum; when positive, you have a minimum.

Small changes (∆x) help us approximate function values without calculating everything from scratch. This saves computation time during exams and real-life applications.

The indefinite integral is the reverse of differentiation. When you differentiate a function, you find its rate of change; integration does the opposite—it finds the original function. Think of it like reversing a journey: if differentiation tells you how fast you're moving, integration tells you the total distance travelled.

The indefinite integral of a function f(x) is written as ∫f(x)dx, and the answer always includes a constant C. For example, if you integrate 2x, you get x² + C, because the derivative of x² is 2x. The constant C matters because many different functions can have the same derivative.

Nigerian businesses use this concept when calculating total profit from a rate of profit function, or finding total revenue from a marginal revenue curve. Learning the standard integration rules like ∫xⁿdx = (xⁿ⁺¹)/(n+1) + C makes solving problems faster.