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Differentiation Cambridge IGCSE Mathematics 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

  1. 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.
  2. dy/dx is the gradient of the curve. To find the gradient at a point, put its x value in.
  3. 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.

  1. Differentiate: dy/dx = 2x − 6.
  2. At a turning point dy/dx = 0: 2x − 6 = 0, so x = 3.
  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.
5 questions, about 2 minutes.

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.

  1. What is dy/dx when y = x⁵?
    • x⁴
    • 5x⁴ (the answer)
    • 5x⁵
    • 4x⁵

    Multiply by 5, and reduce the power to 4.

  2. What is dy/dx when y = 3x²?
    • 3x
    • 5x
    • 6x (the answer)
    • 6x²

    3 × 2 = 6, and x² becomes x.

  3. What is dy/dx when y = 7x?
    • 0
    • 7 (the answer)
    • 7x
    • x

    A straight line y = 7x has gradient 7.

  4. What is dy/dx when y = 9?
    • 0 (the answer)
    • 9
    • 9x
    • 1

    A constant has no slope.

  5. What is the gradient of y = x² + 2x at x = 1?
    • 2
    • 3
    • 4 (the answer)
    • 5

    dy/dx = 2x + 2.

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