Additional Maths questions often combine several familiar steps. A calculus question may also require expanding brackets, solving a quadratic and substituting a coordinate. If your final answer is wrong, the source may be the algebra rather than the calculus rule. Keep those steps separate so you can identify what needs more practice.
Use your own syllabus as the boundary
For IGCSE 0606 exams in 2025–2027, check the official Additional Mathematics syllabus. O Level students should work from their own syllabus code and exam-year papers. The examples below teach shared polynomial-calculus skills; they are a starting point, not a complete content checklist or a forecast of likely questions.
Worked differentiation example
Let y = 3x³ − 5x² + 4x − 7. Differentiate term by term: dy/dx = 9x² − 10x + 4. At x = 2, the gradient is 36 − 20 + 4 = 20. The original curve gives y = 24 − 20 + 8 − 7 = 5, so the corresponding point is (2, 5). The gradient and the y-coordinate are different quantities.
The tangent at that point is y − 5 = 20(x − 2), or y = 20x − 35. Check that the line passes through (2, 5). If a question asks for a normal instead, its gradient is the negative reciprocal of the non-zero tangent gradient: here −1/20. Do not reuse the tangent gradient just because you have already calculated it.
Find and classify a stationary point
For y = x² − 6x + 5, dy/dx = 2x − 6. Set the derivative equal to zero to find x = 3, then substitute into the original equation: y = −4. The stationary point is (3, −4). Since the second derivative is 2, which is positive, this is a minimum. Completing the square gives y = (x − 3)² − 4, providing a second check.
Worked integration example
An antiderivative of 6x² − 4x + 3 is 2x³ − 2x² + 3x + C. For these polynomial terms, increase each power by one and divide by the new power. Differentiate your answer to check that it returns 6x² − 4x + 3. The simple power rule in this form does not apply when the original exponent is −1.
For the definite integral from x = 0 to x = 2, use F(2) − F(0): (16 − 8 + 6) − 0 = 14. The function is positive throughout this interval, so 14 is also the area between the curve and the x-axis over these limits. When a curve crosses the axis, split the region if you need total geometric area rather than signed area.
Diagnose the kind of mistake
| Wrong result | Likely check | Practice task |
|---|---|---|
| Derivative has the wrong power | Apply the rule to one term at a time | Differentiate five separate polynomial terms |
| Correct derivative, wrong coordinate | Substitute into the original function | Pair each gradient with its point |
| Integral fails the derivative check | Check the new coefficient and exponent | Differentiate every antiderivative |
| Negative value for total area | Look for regions below the axis | Sketch before selecting intervals |
Build a focused weekly revision routine
Use one session for algebra prerequisites, one for derivatives and tangents, and one for integrals and area. Finish with a mixed set in which the method is not announced in the heading. Write down why you selected each operation. If you need a tutor to identify the missing step, explore Additional Mathematics lessons. For preparation beyond this course, see our AS to A2 study guide.
Frequently asked questions
01What is the difference between differentiation and integration?
Differentiation finds a rate of change, such as a curve's gradient. Indefinite integration finds a family of antiderivatives. A definite integral can represent signed area or accumulated change, depending on the context.
02Do I always add C when integrating?
Include a constant of integration for an indefinite integral. For a definite integral, evaluate an antiderivative at the two limits and subtract; the constant cancels.
03Does dy/dx = 0 always identify a maximum?
No. It identifies a stationary point when the derivative exists there. Classify the point using an appropriate derivative or sign-change test, and remember that some stationary points are neither maxima nor minima.
