All subjects · Mathematics · WMA13 (P3)

The product rule

Use the product rule to differentiate two functions multiplied together, such as x2 e3x or 2x4 sin x.

The rule

If y = uv, then dy/dx = u dv/dx + v du/dx.

Write down u, v, du/dx and dv/dx first, then substitute. This layout avoids most errors.

Factorising

Answers usually share a factor. x2 e3x gives 2xe3x + 3x2 e3x = xe3x(2 + 3x). Factorised answers make stationary points easy: e3x is never 0.

Combining rules

u or v may themselves need the chain rule, for example x(2x - 1)4 or x3 ln x. Do each derivative carefully before combining.

Worked example

Differentiate y = x2 e3x.

  1. u = x2, du/dx = 2x
  2. v = e3x, dv/dx = 3e3x
  3. dy/dx = x2(3e3x) + e3x(2x)

Answer: dy/dx = xe3x(3x + 2)

Key idea: y = uv gives dy/dx = u v' + v u'. Write u, v, u', v' first and factorise the answer.

Check yourself

  1. Differentiate x sin x.
    • cos x
    • x cos x
    • sin x + x cos x
    • sin x - x cos x
    Show answer

    sin x + x cos x. u = x, v = sin x: x cos x + sin x × 1.

  2. Find the gradient of y = x3 ln x at x = 2, to 3 significant figures.
    Show answer

    12.3. dy/dx = 3x2 ln x + x3(1/x) = 3x2 ln x + x2. At x = 2: 12 ln 2 + 4 = 12.3.

  3. Find the x-coordinate of the stationary point of y = (x + 1)e2x.
    Show answer

    -1.5. dy/dx = e2x + 2(x + 1)e2x = e2x(2x + 3). e2x is never 0, so x = -3/2.

  4. y = 2x4 sin x. Find dy/dx.
    • 8x3 cos x
    • 8x3 sin x + 2x4 cos x
    • 8x3 sin x - 2x4 cos x
    • 2x4 cos x
    Show answer

    8x3 sin x + 2x4 cos x. u = 2x4, v = sin x: u'v + uv' = 8x3 sin x + 2x4 cos x.

  5. Find the gradient of y = x(2x - 1)4 at x = 1.
    Show answer

    9. dy/dx = (2x - 1)4 + x × 8(2x - 1)3. At x = 1: 1 + 8 = 9.

  6. Differentiate x2 e-x, giving your answer factorised.
    • xe-x(2 + x)
    • xe-x(2 - x)
    • -2xe-x
    • x2 e-x(2 - x)
    Show answer

    xe-x(2 - x). 2xe-x - x2 e-x = xe-x(2 - x).

  7. Find the gradient of y = ex cos x at x = 0.
    Show answer

    1. dy/dx = ex cos x - ex sin x = ex(cos x - sin x). At x = 0: 1.

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Last updated 7 November 2026. Spotted a mistake? Tell us.