Implicit differentiation

Terms in y

Differentiate a term in y with respect to y, then multiply by dy/dx (the chain rule):

• d/dx (yⁿ) = ny^{n - 1} dy/dx
• d/dx (e^2y) = 2e^2y dy/dx
• d/dx (sin y) = cos y dy/dx

Products of x and y

Use the product rule: d/dx (xy) = x dy/dx + y. For 3xy²: 3y² + 6xy dy/dx.

Method

1. Differentiate every term (constants give 0).

2. Collect the dy/dx terms on one side.

3. Factorise and divide.

Substitute a point to get a gradient. For stationary points, set the numerator of dy/dx to 0 and solve with the curve's equation.

Worked example

Find the gradient of x² + y² = 25 at (3, 4).

1. 2x + 2y dy/dx = 0
2. dy/dx = -x/y

Answer: dy/dx = -3/4

Check yourself

1. Differentiate y³ with respect to x.
   • 3y²
   • 3x²
   • y³ dy/dx
   • 3y² dy/dx
   Show answer

   3y² dy/dx. Differentiate with respect to y, then multiply by dy/dx.

2. Use the product rule to find d/dx (xy).
   • dy/dx
   • x dy/dx + y
   • x + y dy/dx
   • y dy/dx
   Show answer

   x dy/dx + y. Product rule: x × dy/dx + y × 1.

3. Find the gradient of x² + 3xy + y² = 11 at (1, 2). Give your answer as a fraction in its simplest form.
   Show answer

   -8/7. 2x + 3y + 3x dy/dx + 2y dy/dx = 0, so dy/dx = -(2x + 3y)/(3x + 2y) = -(2 + 6)/(3 + 4) = -8/7.

4. Find dy/dx for x³ + y³ = 9 at the point (1, 2). Give your answer as a fraction in its simplest form.
   Show answer

   -1/4. 3x² + 3y² dy/dx = 0, so dy/dx = -x²/y² = -1/4.

5. Find the gradient of the normal to 4x² + y² = 20 at (2, -2). Give your answer as a fraction in its simplest form.
   Show answer

   -1/4. 8x + 2y dy/dx = 0 gives dy/dx = -4x/y = -8/(-2) = 4. The normal gradient is -1/4.

6. Find the y-coordinates of the stationary points of x² + xy + y² = 12.
   Show answer

   y = -4 or y = 4. dy/dx = -(2x + y)/(x + 2y) = 0 when y = -2x. Then x² - 2x² + 4x² = 12, so x = ±2 and y = ∓4.

7. y is a function of x. What is the derivative of e^2y with respect to x?
   • e^2y dy/dx
   • 2e^2y
   • 2e^2y dy/dx
   • 2e^2x
   Show answer

   2e^2y dy/dx. Chain rule: 2e^2y × dy/dx.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 4 scheme of work

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