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Differentiation: standard results and the chain rule Cambridge IGCSE Additional Mathematics revision

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In plain words

Differentiating finds the gradient of a curve: how fast y is changing as x changes. A short list of standard results, and one rule for functions inside functions, covers almost everything you will be asked to differentiate.

8 things to know

  1. The derivative is written dy/dx or f′(x). Differentiating again gives the second derivative, d²y/dx² or f″(x).
  2. Powers: xⁿ becomes nxⁿ⁻¹, for any rational n. A constant becomes 0.
  3. sin x becomes cos x. cos x becomes −sin x. tan x becomes sec²x. Angles must be in radians.
  4. eˣ stays as eˣ. ln x becomes 1/x.
  5. Sums are differentiated term by term, and a constant multiplier stays in front.
  6. The chain rule, for a function of a function: differentiate the outside, leaving the inside alone, then multiply by the derivative of the inside.
  7. So (ax + b)ⁿ becomes an(ax + b)ⁿ⁻¹, e^(ax) becomes ae^(ax), ln(ax + b) becomes a/(ax + b), and sin(ax) becomes a cos(ax).
  8. No calculus results are given on the formula list. They have to be learned.

Worked example

Differentiate y = (x² + 1)⁴.

  1. The outside is "something to the power 4". Its derivative is 4 × (something)³.
  2. The inside is x² + 1, whose derivative is 2x.
  3. dy/dx = 4(x² + 1)³ × 2x = 8x(x² + 1)³.

Worked example

Differentiate y = e^(3x) + ln(2x + 1) + 5 sin 2x.

  1. e^(3x) becomes 3e^(3x).
  2. ln(2x + 1) becomes 2/(2x + 1).
  3. 5 sin 2x becomes 10 cos 2x. So dy/dx = 3e^(3x) + 2/(2x + 1) + 10 cos 2x.

Tips and tricks

  • Rewrite roots and fractions as powers before differentiating: √x is x^(1/2), and 1/x² is x⁻².
  • The usual chain rule slip is forgetting to multiply by the derivative of the inside. Say it to yourself each time: "times the derivative of the bracket".
7 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Differentiation: standard results and the chain rule: 7 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 the derivative of x⁵?
    • x⁴
    • 5x⁴ (the answer)
    • 5x⁵
    • x⁶/6

    Multiply by the power, then take one off the power.

  2. What is the derivative of cos x?
    • sin x
    • −sin x (the answer)
    • −cos x
    • sec²x

    Note the minus sign.

  3. What is the derivative of e^(4x)?
    • e^(4x)
    • 4e^(4x) (the answer)
    • 4xe^(4x)
    • e^(4x)/4

    The chain rule multiplies by 4.

  4. What is the derivative of ln x?
    • ln x
    • x
    • 1/x (the answer)
    • eˣ

    A standard result.

  5. What is the derivative of (x + 3)⁴?
    • 4(x + 3)⁴
    • 4(x + 3)³ (the answer)
    • (x + 3)³
    • 4x³

    The derivative of the bracket is 1.

  6. What is the derivative of tan x?
    • cot x
    • sec x
    • sec²x (the answer)
    • −cosec²x

    A standard result.

  7. What is the derivative of sin 3x?
    • cos 3x
    • 3 cos 3x (the answer)
    • −3 cos 3x
    • 3 sin 3x

    The chain rule multiplies by 3.

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