Differentiation in SPM Additional Mathematics: From First Principles to A+
Differentiation is one of the highest-weightage topics in SPM Additional Mathematics. Almost every Paper 2 contains at least one full question on it, and it underpins later topics like integration, kinematics, and maximum or minimum problems. This guide breaks the topic into the exact skills examiners test, in the order you should learn them.
What Differentiation Actually Means
Differentiation is the mathematics of rate of change. If a function y = f(x) describes a curve, its derivative, written as dy/dx or f'(x), tells you the gradient of that curve at any point. A steep positive gradient means y is rising fast; a flat zero gradient means a turning point; a negative gradient means y is falling. Once you internalise this single idea, every differentiation question becomes a question about gradients.
The Core Rules You Must Memorise
The power rule is the foundation: if y = axⁿ, then dy/dx = a·n·xⁿ⁻¹. From this, plus the sum, product, quotient, and chain rules, you can differentiate almost any SPM-level function.
| Function | Derivative | Notes |
|---|---|---|
| y = axⁿ | dy/dx = a·n·xⁿ⁻¹ | Works for fractions and negatives too |
| y = sin x | dy/dx = cos x | Angle in radians |
| y = cos x | dy/dx = -sin x | Note the minus sign |
| y = tan x | dy/dx = sec² x | Derived from quotient rule |
| y = eˣ | dy/dx = eˣ | Self-derivative |
| y = ln x | dy/dx = 1/x | Only for x > 0 |
For composite functions like y = (3x + 2)⁵, apply the chain rule: differentiate the outer power, then multiply by the derivative of the inside. Here dy/dx = 5(3x + 2)⁴ · 3 = 15(3x + 2)⁴. Practise this until it is automatic — the chain rule appears in nearly every difficult SPM differentiation question.
Tangents and Normals
A tangent touches the curve at one point and shares the gradient of the curve there. A normal is perpendicular to the tangent, so its gradient is the negative reciprocal. If the tangent's gradient is m, the normal's gradient is -1/m. To find the equation of either line, evaluate dy/dx at the given x-value to get the gradient, then plug the point and gradient into y - y₁ = m(x - x₁).
A common trap: students find dy/dx correctly but forget to substitute the actual x-coordinate. Always evaluate the derivative at the specific point, not leave it as a formula.
Second Derivatives and Turning Points
Differentiating twice gives d²y/dx², the rate of change of the gradient. This is how you classify turning points. Set dy/dx = 0 to locate stationary points, then test them: if d²y/dx² < 0, the point is a maximum; if d²y/dx² > 0, it is a minimum; if it equals zero, the test fails and you must inspect the gradient on either side.
Rates of Change and Connected Rates
Connected rates of change questions link two variables through a third. For example, if the radius of a balloon increases at 2 cm/s, how fast is the volume increasing? Use the chain rule with time: dV/dt = dV/dr · dr/dt. First write V in terms of r, differentiate to get dV/dr, then multiply by the given dr/dt.
Common Mistakes That Cost Marks
- Forgetting to convert degrees to radians before differentiating trig functions.
- Applying the power rule to a negative exponent and forgetting to adjust the sign.
- Confusing dy/dx with the actual gradient at a point — always substitute.
- In connected rates, differentiating the wrong variable or missing the chain-link step.
- Dropping the negative sign on the derivative of cos x.
How to Practise Effectively
Do not move on from a subtopic until you can solve ten consecutive questions without error. Start with past SPM papers (the 2018-2024 papers are excellent), then supplement with topical workbooks. Time yourself on full Paper 2 questions, because under exam pressure, simple sign errors multiply.
Differentiation rewards methodical working. Write each step, label your derivatives, and always finish with a clear final answer in the requested form. That discipline alone will lift you from a B to an A+.