IGCSE Maths / Calculus
Differentiating powers of x
Edexcel International GCSE Maths (4MA1), chapter 28: Calculus, specification point 3.4b. Free lesson.
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Differentiation finds the gradient of a curve at any value of x. For whole number powers, use one rule on each term: multiply by the power, then reduce the power by 1.
How to differentiate a power of x
The derivative, written dy/dx, gives the gradient of a curve at any x. It is an expression because the gradient can change along the curve. For a term xⁿ, the power n is the old power: it comes down as a multiplier before the power is reduced. The rule is dy/dx = nxⁿ⁻¹. For example, x³ becomes 3x², not 2x². The 3 comes down first, then the power drops to 2.
How to differentiate a whole function
Differentiate each term separately, working from left to right. Keep every plus or minus sign in place. This lets you use the same rule even when a function has several different powers.
How to find where a curve is flat
A flat part of a curve has gradient 0. Set dy/dx equal to 0 and solve for x. Then substitute each x value into the original function to find the full coordinates. For y = x³ − 3x² − 9x + 5, set 3x² − 6x − 9 = 0. Factorising gives 3(x + 1)(x − 3) = 0, so x = −1 or x = 3. Substituting into the original function gives the points (−1, 10) and (3, −22). Both are points where the curve is flat.
What happens to a plain number or a plain x
A number on its own has no x in it. Its graph is a horizontal line, so its gradient is 0 and it disappears when differentiated. A term with x behaves differently, even when its power is only 1. For −9x, write the power as 1: −9x = −9x¹. The 1 comes down and the power becomes 0, giving −9x⁰ = −9. So a plain x leaves its coefficient behind, but a plain number differentiates to 0.
Check the signs and recognise differentiation questions
Differentiation questions may ask for a gradient, a rate of change, or the position of a maximum or minimum. These all involve the derivative. The differentiation rule is not on the formula sheet, so learn it and apply it term by term. For y = 4x³ − 7x² + 2x − 6, the derivative is 12x² − 14x + 2. Keep the original signs: −7x² gives −14x, and −6 disappears.