Differentiation Cambridge IGCSE Mathematics (9–1) revision
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In plain words
The gradient of a straight line is the same everywhere. A curve's gradient keeps changing: it might be steep here and flat there. Differentiation gives a formula for the gradient at any point, and that tells you where the curve's peaks and dips are.
Three things to know
- To differentiate a power of x, multiply by the power and take one off it: xⁿ becomes nxⁿ⁻¹. So x³ becomes 3x², 5x² becomes 10x, 7x becomes 7, and a number on its own becomes 0.
- dy/dx is the gradient of the curve. To find the gradient at a point, put its x value in.
- At a turning point (a maximum or a minimum) the gradient is zero. So solve dy/dx = 0.
Worked example
Find the turning point of y = x² − 6x + 1.
- Differentiate: dy/dx = 2x − 6.
- At a turning point dy/dx = 0: 2x − 6 = 0, so x = 3.
- Put x = 3 into the equation of the curve: y = 9 − 18 + 1 = −8. The turning point is (3, −8).
Tips and tricks
- Differentiate each term separately, one at a time.
- After solving dy/dx = 0, go back to the original equation for y, not to dy/dx.
It lands in your notebook with its questions as flashcards.
Differentiation: 5 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
What is dy/dx when y = x⁵?
Multiply by 5, and reduce the power to 4.
What is dy/dx when y = 3x²?
3 × 2 = 6, and x² becomes x.
What is dy/dx when y = 7x?
A straight line y = 7x has gradient 7.
What is dy/dx when y = 9?
A constant has no slope.
What is the gradient of y = x² + 2x at x = 1?
dy/dx = 2x + 2.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
4 questions, 10 marks. Write your answers on paper, then check them.
Differentiation
Cambridge IGCSE Mathematics (9–1) 0980 · 10 marks · papermunch.org
Name ______________________________ Date ______________
Differentiate y = 4x³.[2]
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dy/dx = 12x².
Differentiate y = x² + 5x − 3.[2]
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dy/dx = 2x + 5.
Find the gradient of the curve y = x² at the point where x = 3.[2]
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6. dy/dx = 2x.
Find the coordinates of the turning points of y = x³ − 3x² + 2.[4]
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(0, 2) and (2, −2). dy/dx = 3x² − 6x = 3x(x − 2) = 0.
Answers: Differentiation
- 1. dy/dx = 12x².
- 2. dy/dx = 2x + 5.
- 3. 6. dy/dx = 2x.
- 4. (0, 2) and (2, −2). dy/dx = 3x² − 6x = 3x(x − 2) = 0.