Differentiating after expanding revision notes

Edexcel International GCSE Maths (4MA1), chapter 28: Calculus, specification point 3.4b. Free lesson.

Watch the video lesson · IGCSE Maths past papers

When a function contains multiplied brackets, expand them into separate powers of x before differentiating. Then collect like terms and differentiate each term on its own.

Why you expand before differentiating

The differentiation rule works on separate powers of x that are added or subtracted. A product of brackets is not in that form, so expand it first. A term is one separate part of the expanded expression, such as x² or −5. If you see multiplied brackets, make the expanded version the first line of your working. A grid can help you make sure every term in one bracket is multiplied by every term in the other.

How to expand two brackets accurately

For (x + 5)(x − 1), each term in the first bracket multiplies each term in the second. Keep the minus sign attached to −1 throughout, then combine the like terms. The minus sign in the second bracket affects both boxes in its column, so check those signs carefully.

How to differentiate and find the lowest point

Once the brackets are gone, differentiate each term. For x² + 4x − 5, the derivative is 2x + 4. Setting the derivative equal to zero finds where the curve is flat. Here, that point is the lowest point. The derivative gives the curve's gradient, which tells you how steep it is. Questions may ask for differentiation using words such as minimum, gradient or rate of change.

How to expand a squared bracket

A squared bracket means the bracket is multiplied by itself. Expand that part first, then multiply by the remaining bracket. Collect like terms before differentiating, so each part of the expression is counted correctly.

What not to do with multiplied brackets

Do not differentiate each bracket separately and multiply the results. There is no rule at this level for differentiating two expressions multiplied together. Expand first, then differentiate the resulting terms. For (x + 5)(x − 1), differentiating each bracket to 1 and writing 1 × 1 = 1 is incorrect. The correct expanded form is x² + 4x − 5, so dy/dx = 2x + 4.

Quick check: a bracket multiplied by x²

Multiplying a single bracket by x² is also an expansion. Multiply x² into each term before differentiating.