Differentiation and Limits

Differentiation is a core branch of calculus that deals with the study of rates at which quantities change. In the JAMB syllabus, this focuses on the limit of functions and the differentiation of algebraic and basic trigonometric functions.

1. The Limit of a Function

The limit describes the behavior of a function as the independent variable approaches a specific value. We denote it as:
lim (x → a) f(x) = L

Methods of finding limits:

• Direct Substitution: Simply plug the value of 'a' into the function.
• Factorization: If substitution results in an indeterminate form like 0/0, factorize the numerator and denominator to cancel out common terms.
• Rationalization: Used when the function contains square roots.
• Limits at Infinity: For rational functions, divide every term by the highest power of x in the denominator.

2. Differentiation from First Principles (Brief)

The derivative of a function f(x) is defined as the limit of the gradient of the secant line as the interval approaches zero:
f'(x) = lim (h → 0) [f(x+h) - f(x)] / h

3. Differentiation of Algebraic Functions

For explicit algebraic functions, we primarily use the Power Rule:

• If y = xⁿ, then dy/dx = nx^n-1
• If y = axⁿ, then dy/dx = anx^n-1
• The derivative of a constant is always 0.
• The derivative of a sum/difference is the sum/difference of the derivatives: d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

4. Differentiation of Trigonometric Functions

JAMB requires knowledge of the derivatives of the three primary trigonometric ratios:

• d/dx(sin x) = cos x
• d/dx(cos x) = -sin x
• d/dx(tan x) = sec²x

Example: Differentiate y = 3x² + 4sin(x).
Solution: dy/dx = 6x + 4cos(x).