The Chain Rule
Simple Explanation
To differentiate a COMPOSITE function — a function wrapped around another function, like f(g(x)) — differentiate the outer function first (leaving the inner function unchanged inside it), then multiply by the derivative of the inner function: d/dx[f(g(x))] = f'(g(x))·g'(x).
Why Do We Need It?
Composite functions are extremely common (e.g. any expression with something other than plain x raised to a power, or inside a root), and the Chain Rule is the only correct way to differentiate them.
See It
A parabola shifted to the right, y equals x minus 2 squared, with a straight tangent line touching the curve at the point (4,4)
Formula
The Chain Rule
d/dx[f(g(x))] = f'(g(x))·g'(x)
To differentiate a composite function (a function of a function), differentiate the outer function first (leaving the inner function unchanged inside it), then multiply by the derivative of the inner function.
- f(g(x))
- — a composite function — f applied to g(x)
- g'(x)
- — the derivative of the inner function g
When to use it: Whenever differentiating a function that is "wrapped around" another function, e.g. (3x+1)^5 or sin(x²).
Worked Example
Apply the Chain Rule
Differentiate f(x) = (x−2)², using the Chain Rule, then find f'(4).
Why Does This Work?
A small change Δx in x causes a corresponding small change Δu in the inner function, which in turn causes a small change Δy in the outer function. The overall rate Δy/Δx can be split as (Δy/Δu)·(Δu/Δx) — and taking the limit as Δx→0 turns this into dy/dx = (dy/du)·(du/dx), the Chain Rule.
Real-Life Example
An expanding oil slick's changing area
A circular oil slick's area depends on its radius (Area = πr²), and the radius itself is growing over time.
The Chain Rule links the two relationships — how area depends on radius, and how radius depends on time — to find how fast the area itself is growing, a classic "related rates" application.
Practice
Differentiate f(x) = (2x−1)³ using the Chain Rule, then evaluate f'(1).
HardCommon mistake
Forgetting to multiply by the inner function's derivative (du/dx) — differentiating only the outer function and stopping there is a very common Chain Rule error.
Quick Review
- d/dx[f(g(x))] = f'(g(x))·g'(x).
- Differentiate the outer function first (keeping the inner function unchanged inside), then multiply by the inner function's derivative.
- Essential whenever one function is "wrapped around" another.