Chain rule
MathDefinition
A differentiation rule for finding the derivative of a composite function. If y = f(g(x)), the derivative is f′(g(x)) · g′(x). In words: take the derivative of the outer function evaluated at the inner function, then multiply by the derivative of the inner function.
How It Works
- Identify the outer function f and the inner function g(x).
- Differentiate the outer function, keeping the inner function unchanged.
- Multiply by the derivative of the inner function.
- Simplify the result.
Examples
- d/dx[sin(3x)] = cos(3x) · 3 = 3cos(3x)
- d/dx[(x² + 1)⁵] = 5(x² + 1)⁴ · 2x = 10x(x² + 1)⁴
Key Fact
d/dx[f(g(x))] = f′(g(x)) · g′(x)
Study This Concept
Practice chain rule with free review games in these units: