Higher Maths worked question

Differentiate a composite power using the chain rule

A fixed, reviewed chain rule question with a complete worked solution and method explanation.

Standard3 marksNon-calculator

Question

Differentiate y = (3x − 1)⁴

Show the worked solution

Worked solution

  1. Let u = 3x − 1, so y = u⁴

    This separates the inner linear function from the outer fourth power.

  2. dy/du = 4u³ and du/dx = 3

  3. dy/dx = (dy/du)(du/dx) = 4u³ × 3 = 12u³

    Apply the chain rule.

  4. dy/dx = 12(3x − 1)³

    Replace u with the original inner expression.

Final answer: dy/dx = 12(3x − 1)³

Why this method works

Differentiate the outer power while keeping the inner expression unchanged, then multiply by the derivative of the inner expression.

Check your work

The power should reduce from 4 to 3, and the coefficient should include the inner derivative 3.

Common mistakes

  • Forgetting to multiply by the inner derivative.
  • Changing the inner expression while differentiating the outer function.
  • Reducing the power without multiplying by 4.