What is Differentiation?
Differentiation is the branch of calculus that studies how quantities change. It gives us the exact rate at which one variable changes with respect to another — the mathematical language of motion, growth, and optimisation.
Consider a car travelling along a straight road. Its position changes over time. The speed at any instant is the rate of change of position with respect to time. Differentiation gives us a precise way to calculate that instantaneous rate — not an average over an interval, but the exact value at a single moment. This idea underpins engineering, economics, physics, biology, and every quantitative science taught at Kenyan universities.
In the CBC Senior Secondary curriculum, differentiation is introduced as part of the Pure Mathematics strand. Students are expected to understand the derivative conceptually, apply the standard rules fluently, and use differentiation to solve practical problems involving rates of change, optimisation, and curve sketching.
The Gradient of a Curve
For a straight line, the gradient (slope) is constant: rise ÷ run. For a curve, the gradient is different at every point. The derivative of a function gives the gradient of the tangent line to the curve at any chosen point.
f′(x) > 0 → curve is rising (increasing) at x
f′(x) < 0 → curve is falling (decreasing) at x
f′(x) = 0 → tangent is horizontal → possible stationary point
Notation for the Derivative
Several equivalent notations are used across different contexts and textbooks:
Differentiation from First Principles
First principles shows where the derivative comes from. Before applying any rules, every student should understand this foundational definition — it is the formal mathematical foundation upon which all differentiation rests.
The idea begins with a secant line — a straight line connecting two points on a curve. If those two points are very close together, the secant line approximates the tangent. As the horizontal distance between the points approaches zero, the secant line becomes the tangent, and its gradient becomes the derivative.
Here, h represents the small horizontal change (sometimes written as Δx or δx). As h approaches 0, we are making the interval smaller and smaller until we arrive at the instantaneous rate of change at exactly the point x.
Basic Differentiation Rules
Once first principles are understood, these four rules allow us to differentiate almost any polynomial function instantly — without repeating the limit process each time.
Applying the Power Rule — Special Cases
The power rule applies not only to positive integer powers but also to negative powers, fractional powers, and roots. The key step is to rewrite the expression in the form xⁿ first.
Derivatives of Trig, Exponential & Log Functions
These are standard results that must be memorised. They are proved using first principles or limits of trigonometric functions, but at secondary level you are expected to know and apply them directly.
Product, Quotient & Chain Rules
These three rules extend differentiation to combinations of functions — products, quotients, and compositions. They are among the most frequently examined rules in both KCSE and CBC Senior Secondary mathematics.
How to Choose Which Rule to Use
Stationary Points, Maxima & Minima
A stationary point is any point where the gradient equals zero — that is, where the tangent line is horizontal. Identifying and classifying stationary points is one of the most important applications of differentiation in KCSE and CBC Senior Secondary assessments.
At x = −1: f′′(−1) = −6 < 0 → local maximum (−1, 4)
y(1) = 2+3−12+1 = −6
At x = 1: 12(1)+6 = 18 > 0 → minimum
Rates of Change & Optimisation
Rates of change and optimisation are the most practically valuable applications of differentiation. They appear in every field from engineering and physics to economics and biology — and in every KCSE Paper 2 examination.
Connected Rates of Change
When two quantities are related through an equation, their rates of change are connected through the chain rule. If y depends on x which depends on t (time), then:
Equations of Tangents & Normals
The derivative gives the gradient of the tangent line to a curve at any point. The normal is perpendicular to the tangent at that same point. Both are regularly examined in KCSE Paper 2.
Implicit & Parametric Differentiation
Not all functions are written as y = f(x). Implicit and parametric forms require specialised differentiation techniques that extend beyond the standard rules.
Implicit Differentiation
An implicit function is one where y cannot easily be isolated — e.g. x² + y² = 25. To differentiate, apply d/dx to both sides. When you differentiate a term involving y, multiply by dy/dx using the chain rule (since y is a function of x).
Parametric Differentiation
In parametric equations, both x and y are expressed as functions of a third variable (the parameter), usually t. To find dy/dx, use the chain rule in the form:
Partial Derivatives & Multivariable Calculus
When a function depends on more than one variable — e.g. temperature depends on both position and time — we need partial derivatives to study how the function changes with respect to each variable independently while holding the others fixed.
The Second Derivative Test for Two Variables
Taylor & Maclaurin Series
Taylor and Maclaurin series express functions as infinite polynomials using their derivatives. This powerful technique allows computers to compute transcendental functions like sin, cos, and eˣ numerically, and underpins much of numerical analysis and physics.
Common Maclaurin Series — Must Know
Complete Differentiation Formula Sheet
Use this as your quick-reference summary for all rules and standard results covered in this lesson.
Common Errors & How to Avoid Them
These are the errors most frequently seen by examiners in KCSE and CBC Senior Secondary marking scripts. Recognising and avoiding these errors can mean the difference between a B and an A.