Higher Maths worked question

Differentiate a power with a quadratic inner function

Practise using the chain rule to differentiate a composite power, then check each stage with a full worked solution.

Challenge3 marksNon-calculator

Question

Differentiate y = (2x2 + 3)4

Show the worked solution

Worked solution

  1. Let u = 2x2 + 3, so y = u4

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

  2. dy/du = 4u3 and du/dx = 4x

    Differentiate the outer power and the inner quadratic separately.

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

    The chain rule multiplies the outer derivative by the inner derivative.

  4. dy/dx = 16x(2x2 + 3)3

    Multiply the coefficients and replace u with 2x2 + 3

Final answer: dy/dx = 16x(2x2 + 3)3

Why this method works

Treat the bracket as one inner function. Differentiating the outer fourth power produces 4(2x2 + 3)3, and the chain rule then requires multiplication by the inner derivative 4x.

Check your work

The outer power must reduce from 4 to 3, and the final derivative must contain the factor 4x from differentiating 2x2 + 3.

Common mistakes

  • Using an inner derivative of 2x instead of 4x.
  • Forgetting the inner derivative entirely.
  • Expanding the fourth power and creating unnecessary algebra.