Mathematics ยท WMA11 (P1)

Differentiating powers of x

Differentiation gives the gradient of a curve at any point. For powers of x there is one simple rule.

The rule

If y = axn, then dy/dx = anxn-1.

Multiply by the power, then reduce the power by 1. This works for any rational n, including negative and fractional powers.

Before you differentiate

Rewrite every term as a power of x first:

Gradient at a point

Substitute the x-coordinate into dy/dx to get the gradient of the curve (and of its tangent) at that point.

Worked example

Find the gradient of y = x3 - 5x + 2 at x = 2.

  1. dy/dx = 3x2 - 5
  2. At x = 2: 3(4) - 5 = 7

Answer: The gradient is 7.

Key idea: y = axn gives dy/dx = anxn-1. Rewrite roots and fractions as powers first.

Check yourself

  1. y = 4x3. Find dy/dx when x = 1.
    Show answer

    12. dy/dx = 12x2 = 12 at x = 1.

  2. What is dy/dx for y = x-2?
    • -2x-1
    • -2x-3
    • 2x-3
    • -x-3
    Show answer

    -2x-3. Multiply by -2 and reduce the power to -3.

  3. How should sqrt(x) be written before differentiating?
    • x2
    • x-1/2
    • 2x
    • x1/2
    Show answer

    x1/2. A square root is the power 1/2.

  4. Find the gradient of y = x2 + 3x at x = 4.
    Show answer

    11. dy/dx = 2x + 3 = 11 at x = 4.

  5. y = 7. What is dy/dx?
    Show answer

    0. A constant has zero gradient.

Practise with instant feedback

Sources

Last updated 7 November 2026. Spotted a mistake? Email alevel@lifereadysociety.com.