The Chain Rule
“Chain rule: outer derivative × inner derivative, multiplied down the chain.”
The formula
(f(g(x)))' = f'(g(x))·g'(x)How to read it: differentiate the outer function (leaving the inside as is), then multiply by the derivative of the inner function
- g(x)
- — the inner function
- f'(g(x))
- — differentiate the outer function, leaving the inside untouched
- g'(x)
- — the derivative of the inner function
The hook
When a function sits inside another function, the derivative flows as 'outer × inner' — the rule you'll reach for most in calculus.
In plain words
To differentiate a composite (a function inside a function), multiply the outer function's derivative by the inner function's derivative.
The intuition
Picture meshed gears. The outer wheel turns at some multiple of the inner wheel's speed. Rates of change multiply down the chain — hence 'chain' rule.
How it's built
Peel it like an onion, outermost first. Differentiate the outer f (leaving the inside g alone), then multiply by how fast g itself changes with x (that's g').
Example
For y=(x²+1)³, the outer is a cube (→3u²), the inner is x²+1 (→2x). Multiply: y'=3(x²+1)²·2x = 6x(x²+1)². At x=1 that's 6·1·4 = 24.
Common mistake
It's easy to forget to multiply by the inner derivative g'. Stopping at 3(x²+1)² — outer only — is wrong.
Where it's used
Whenever exponentials, logs, or trig functions are wrapped around other expressions; related rates in physics; and backpropagation in neural networks — all chain rule.
Where it came from
This is where Leibniz's dy/dx notation shines — dy/dx = dy/du · du/dx reads as if the fractions just cancel.
Prerequisites
Quick check
What is the derivative of y=(3x+1)²?
- 2(3x+1)
- 6(3x+1)✓
- 6x
- 3(3x+1)²
Practice
What is the derivative of y=(2x+1)³ at x=0?
Answer: 6
- Outer: cube → 3(2x+1)² ; inner: 2x+1 → 2
- y'=3(2x+1)²·2 = 6(2x+1)²
- Plug in x=0 → 6·1 = 6
Takeaway: After the outer derivative, always multiply by the inner derivative 2.
What is the derivative of y=(x²+1)² at x=1?
Answer: 8
- Outer: square → 2(x²+1) ; inner: x²+1 → 2x
- y'=2(x²+1)·2x = 4x(x²+1)
- Plug in x=1 → 4·1·2 = 8
Takeaway: Expand to x⁴+2x²+1, differentiate to 4x³+4x, at x=1 gives 8 — matches.
Differentiate y=(5x−2)⁴ with the chain rule.
Answer: undefined
- Outer: 4th power → 4(5x−2)³ ; inner: 5x−2 → 5
- y'=4(5x−2)³·5 = 20(5x−2)³
Takeaway: Don't forget to multiply by the inner derivative (here, 5).