GES Elective Mathematics: Calculus and
Differentiation
1. Introduction to Calculus
Calculus is the branch of mathematics that studies change. It is divided into:
- Differential Calculus – concerned with the rate of change (derivatives).
- Integral Calculus – concerned with accumulation of quantities (integrals).
We’ll focus on Differentiation in this module.
2. Concept of Limits
The limit of a function f(x) as x → a is the value that f(x) approaches as x gets close to a.
Notation:
lim(x → a) f(x) = L
Example:
lim(x → 2) (3x + 1) = 7
3. Definition of Derivative
The derivative of a function f(x) is defined as:
f'(x) = lim(h → 0) [(f(x+h) - f(x))/h]
This gives the slope of the tangent to the curve at a point.
Geometric Interpretation: The derivative represents the gradient of the curve y = f(x) at any
point x.
4. Rules of Differentiation
• Power Rule: d/dx (x^n) = nx^(n-1)
• Constant Rule: d/dx (c) = 0
• Constant Multiple Rule: d/dx [c * f(x)] = c * f'(x)
• Sum and Difference Rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x)
• Product Rule: d/dx [f(x) * g(x)] = f'(x)g(x) + f(x)g'(x)
• Quotient Rule: d/dx [f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2
• Chain Rule: dy/dx = (dy/du) * (du/dx)
