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MathidiaMathematics Academy
Advanced differentiation
Lesson 28 of 75
Standard practice

Chain rule

Objective. Differentiate nested algebraic, exponential and trigonometric functions.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

The chain rule differentiates a composite function, one function nested inside another. If y = f(g(x)), then dy/dx = f′(g(x)) × g′(x): differentiate the outer function leaving the inner function unchanged, then multiply by the derivative of the inner function. For example, if y = sin(3x²), the outer function is sine and the inner is 3x², so dy/dx = cos(3x²) × 6x = 6x cos(3x²). Spotting the outer and inner layers first makes the method reliable for exponential and trigonometric composites too.

Worked example

Differentiate y = e^(5x³) using the chain rule.

5
Coordinate grid with the line 5
  1. 1Identify outer function e^u and inner function u = 5x³
  2. 2Differentiate the outer function, keeping the inner unchanged: e^(5x³)
  3. 3Differentiate the inner function: du/dx = 15x²
  4. 4Multiply the two results: dy/dx = 15x²e^(5x³)

Answer: dy/dx = 15x²e^(5x³)

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Common mistake. A frequent slip is forgetting to multiply by the derivative of the inner function, writing dy/dx = cos(3x²) alone. Always finish by multiplying by the inner function's own derivative.

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