Differentiation Revision Notes for IGCSE Maths
These comprehensive revision notes cover everything you need to know about differentiation for the Cambridge IGCSE Mathematics 0580 examination. Written by Teacher Rig, each section includes key concepts, essential formulas, and practical exam tips to help you achieve your best grade.
The Power Rule
Differentiation finds the gradient function (rate of change). For y = ax^n, the derivative is dy/dx = nax^(n-1). Multiply by the power, then reduce the power by 1. Constants differentiate to zero. The derivative represents the gradient of the tangent at any point on the curve.
Key Formulas
- If y = ax^n, then dy/dx = nax^(n-1)
- If y = c (constant), then dy/dx = 0
- If y = f(x) + g(x), then dy/dx = f'(x) + g'(x)
Exam Tips
- Rewrite fractions and roots as powers before differentiating: 1/x = x^(-1), sqrt(x) = x^(1/2)
- Differentiate each term independently
- The derivative of a constant is always zero
Finding the Gradient at a Point
The gradient of a curve at a specific point equals the value of the derivative at that point. Differentiate to find dy/dx, then substitute the x-value.
Key Formulas
- Gradient at x = a is dy/dx evaluated at x = a
Exam Tips
- The gradient of the TANGENT equals the gradient of the CURVE at that point
- The gradient of the NORMAL is the negative reciprocal of the tangent gradient
- A positive gradient means the function is increasing, negative means decreasing
Stationary Points
Stationary points occur where dy/dx = 0. To classify: find the second derivative d2y/dx2. If d2y/dx2 > 0, it is a minimum. If d2y/dx2 < 0, it is a maximum. If d2y/dx2 = 0, the test is inconclusive and you need to check the gradient on either side.
Key Formulas
- Stationary point: dy/dx = 0
- Maximum: d2y/dx2 < 0
- Minimum: d2y/dx2 > 0
Exam Tips
- Setting dy/dx = 0 is worth a mark on its own - always write this step
- Find y-coordinates by substituting back into the ORIGINAL equation, not the derivative
- Questions often ask you to 'determine the nature' - this means use the second derivative
Tangents and Normals
The tangent to a curve at a point touches the curve and has the same gradient as the curve at that point. The normal is perpendicular to the tangent at that point. Use y - y1 = m(x - x1) to find the equation of either.
Key Formulas
- Tangent gradient = dy/dx at the point
- Normal gradient = -1 / (dy/dx at the point)
- Line equation: y - y1 = m(x - x1)
Exam Tips
- Find the y-coordinate first, then the gradient, then use the line equation formula
- For the normal, use the NEGATIVE RECIPROCAL of the tangent gradient
- These questions often combine with other topics like solving simultaneous equations
Revision Checklist
- I understand all key concepts in differentiation
- I have memorised the essential differentiation formulas
- I can apply these concepts to exam-style questions
- I have practised past paper questions on differentiation
- I know the common mistakes to avoid in differentiation questions
Frequently Asked Questions
What differentiation topics are covered in IGCSE Maths?
The IGCSE 0580 syllabus covers differentiation across both Core and Extended tiers. Key areas include the power rule. Key areas include finding the gradient at a point. Key areas include stationary points.
How important is differentiation in the IGCSE exam?
Differentiation is a significant part of the IGCSE Mathematics exam, typically appearing in Paper 2 (non-calculator) and Paper 4 (calculator). Questions range from straightforward calculations to multi-step problems that combine differentiation with other topics.
What are the most common mistakes in differentiation?
Common mistakes include not showing full working, forgetting to state units, misreading the question, and rushing through calculations. For differentiation specifically, make sure you understand the underlying concepts rather than just memorising procedures.
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