Chain Rule Formula
Apply the chain rule for derivatives: d/dx[f(g(x))] = f'(g(x)) times g'(x).
Essential calculus technique for composite functions.
The Formula
The chain rule is one of the most important differentiation techniques in calculus. It tells you how to find the derivative of a composite function — a function built by plugging one function inside another. Without the chain rule, you would be unable to differentiate most real-world functions, since they almost always involve compositions.
The idea behind the chain rule is intuitive: if you want to know how fast a composite function changes, you multiply the rate of change of the outer function by the rate of change of the inner function. Think of it like gears in a machine. If the inner gear turns twice as fast as the input, and the outer gear turns three times as fast as the inner gear, then the output turns six times as fast as the input.
Gottfried Wilhelm Leibniz developed an early form of the chain rule in the late 1600s in Germany as part of his invention of calculus. The modern notation and formal statement were refined over the following centuries. Today, the chain rule is typically one of the first derivative rules taught in any calculus course, right after the power rule, product rule, and quotient rule.
In practice, you identify the outer function f and the inner function g(x), differentiate each separately, and then multiply the results together. The key step is remembering to evaluate the derivative of the outer function at the inner function, not at x directly. This is where many students make errors.
The chain rule also extends to multiple compositions. If you have f(g(h(x))), you take f'(g(h(x))) × g'(h(x)) × h'(x). Each layer peels off one derivative, evaluated at all the functions still inside it.
Variables
| Symbol | Meaning |
|---|---|
| f(g(x)) | 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 with respect to x |
| d/dx | Differentiation with respect to x |
Example 1
Problem
Find the derivative of y = (3x + 1)5.
Outer function: f(u) = u5, so f'(u) = 5u4
Inner function: g(x) = 3x + 1, so g'(x) = 3
Apply chain rule: dy/dx = 5(3x + 1)4 × 3
dy/dx = 15(3x + 1)4
Example 2
Problem
Find the derivative of y = sin(x2).
Outer function: f(u) = sin(u), so f'(u) = cos(u)
Inner function: g(x) = x2, so g'(x) = 2x
Apply chain rule: dy/dx = cos(x2) × 2x
dy/dx = 2x cos(x2)
When to Use It
The chain rule is needed whenever you differentiate a composite function — which is most functions in practice.
- Differentiating powers of expressions, like (2x + 5)10
- Finding derivatives of trigonometric functions with non-trivial arguments, like sin(3x) or cos(x2)
- Differentiating exponential and logarithmic compositions, like e2x or ln(x2 + 1)
- Implicit differentiation and related rates problems in applied calculus