Chain Rule
Reference for the chain rule d/dx[f(g(x))] = f'(g(x))*g'(x).
Step-by-step examples with trig, exponential, and nested polynomial composite functions.
The Chain Rule
dy/dx = f'(g(x)) · g'(x)
The chain rule is used to differentiate composite functions — a function inside another function. Think of it as peeling layers: differentiate the outer function first, then multiply by the derivative of the inner function.
In Leibniz Notation
dy/dx = (dy/du) · (du/dx)
Variables
| Symbol | Meaning |
|---|---|
| f(g(x)) | A composite function (outer function f applied to inner function g) |
| f'(g(x)) | Derivative of the outer function, evaluated at g(x) |
| g'(x) | Derivative of the inner function |
Example 1 — Simple Chain Rule
Find d/dx [(3x + 2)⁵]
Outer function: u⁵, Inner function: u = 3x + 2
d/dx [u⁵] = 5u⁴ · du/dx
= 5(3x + 2)⁴ · 3
= 15(3x + 2)⁴
Example 2 — Trigonometric Chain Rule
Find d/dx [sin(x²)]
Outer function: sin(u), Inner function: u = x²
= cos(x²) · d/dx[x²]
= 2x · cos(x²)
Example 3 — Exponential Chain Rule
Find d/dx [e^(4x³)]
Outer function: e^u, Inner function: u = 4x³
= e^(4x³) · d/dx[4x³]
= 12x² · e^(4x³)
Example 4 — Double Chain Rule
Find d/dx [sin²(3x)]
This is [sin(3x)]². Three layers: square → sin → 3x
= 2·sin(3x) · d/dx[sin(3x)]
= 2·sin(3x) · cos(3x) · 3
= 6·sin(3x)·cos(3x) = 3·sin(6x)
When to Use It
Use the chain rule whenever you see:
- A function raised to a power: (something)ⁿ
- A trig function of an expression: sin(something), cos(something)
- An exponential with a non-trivial exponent: e^(something)
- A logarithm of an expression: ln(something)
- Any nested or composite function
Key Notes
- Formula: d/dx f(g(x)) = f'(g(x)) × g'(x): Differentiate the outer function (leaving the inner function unchanged), then multiply by the derivative of the inner function. This must be applied once for each layer of composition.
- Leibniz notation: dy/dx = (dy/du)(du/dx): If y = f(u) and u = g(x), then dy/dx equals the product of the two intermediate rates. While not literally fraction cancellation, this notation makes the chain rule intuitive and helps organize multi-layer derivatives.
- Multiple layers — work from outside in: For h(x) = sin(e^(x²)), differentiate outermost first: cos(e^(x²)) × d/dx[e^(x²)] = cos(e^(x²)) × e^(x²) × 2x. Each layer contributes one factor to the product.
- Implicit differentiation uses the chain rule: When differentiating y³ with respect to x, treat y as a function of x: d/dx[y³] = 3y² × dy/dx. The extra dy/dx factor is the chain rule applied to the implicit function y(x).
- Applications: The chain rule is used whenever composite functions arise — virtually all of applied calculus. It underlies backpropagation in neural networks (the gradient of a loss with respect to weights through layers of functions), related rates problems, and most physical equations involving coupled variables.