The chain rule

The rule

If y is a function of u, and u is a function of x, then dy/dx = dy/du × du/dx.

In words: differentiate the outside, keep the inside, then multiply by the derivative of the inside.

Patterns to know

• d/dx (f(x))ⁿ = n f'(x)(f(x))^{n - 1}
• d/dx e^(f(x)) = f'(x) e^(f(x))
• d/dx ln f(x) = f'(x)/f(x)
• d/dx sin f(x) = f'(x) cos f(x)

tan² 2x means (tan 2x)², so it needs the chain rule twice.

dy/dx = 1/(dx/dy)

If x is given in terms of y, such as x = y² + 3y or x = sin³ y, find dx/dy and take its reciprocal. You may then need to rewrite the answer in terms of x.

Worked example

Differentiate y = (3x² + 1)⁵.

1. Let u = 3x² + 1, so y = u⁵.
2. dy/du = 5u⁴ and du/dx = 6x.
3. dy/dx = 5u⁴ × 6x

Answer: dy/dx = 30x(3x² + 1)⁴

Check yourself

1. Differentiate (2x - 1)⁶.
   • 6(2x - 1)⁵
   • 12(2x - 1)⁵
   • 12(2x - 1)⁶
   • 6(2)⁵
   Show answer

   12(2x - 1)⁵. 6(2x - 1)⁵ × 2 = 12(2x - 1)⁵.

2. Find the gradient of y = sqrt(4x + 1) at x = 2. Give your answer as a fraction in its simplest form.
   Show answer

   2/3. y = (4x + 1)^1/2, so dy/dx = (1/2)(4x + 1)^-1/2 × 4 = 2/sqrt(4x + 1). At x = 2: 2/3.

3. Differentiate ln(x² + 1).
   • 1/(x² + 1)
   • 2x/(x² + 1)
   • 2x ln(x² + 1)
   • 1/(2x)
   Show answer

   2x/(x² + 1). d/dx ln f(x) = f'(x)/f(x), with f'(x) = 2x.

4. Find the gradient of y = e^{x2 - 4x} at x = 1, to 3 significant figures.
   Show answer

   -0.0996. dy/dx = (2x - 4)e^{x2 - 4x}. At x = 1: -2e^-3 = -0.0996.

5. Differentiate cos(x²).
   • -2x sin(x²)
   • -sin(x²)
   • -sin(2x)
   • 2x sin(x²)
   Show answer

   -2x sin(x²). The outside is cos, giving -sin(x²); the inside x² gives 2x.

6. Differentiate tan² 2x.
   • 2tan 2x sec² 2x
   • 4sec² 2x
   • 4tan 2x sec² 2x
   • 2sec⁴ 2x
   Show answer

   4tan 2x sec² 2x. tan² 2x = (tan 2x)². d/dx = 2tan 2x × 2sec² 2x.

7. x = y² + 3y. Find dy/dx when y = 1. Give your answer as a fraction in its simplest form.
   Show answer

   1/5. dx/dy = 2y + 3 = 5, so dy/dx = 1/5.

8. x = e^2y. Find dy/dx in terms of x.
   • 2x
   • 1/(2x)
   • 1/x
   • 2e^2x
   Show answer

   1/(2x). dx/dy = 2e^2y = 2x, so dy/dx = 1/(2x).

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 3 scheme of work

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