Differentiate (3x + 1)² by spotting its inner and outer functions.
A function inside a function
In (3x + 1)², first you calculate 3x + 1. Then you square that result. There are two layers: the inner function g(x) = 3x + 1 and the outer function f(u) = u².
Take it one layer at a time
The derivative of the outer function u² is 2u.
Put the inner function back: 2(3x + 1).
The derivative of the inner function 3x + 1 is 3.
Multiply: 2(3x + 1) × 3 = 6(3x + 1).
In symbols, when f and g are differentiable, the derivative of f(g(x)) is f′(g(x)) × g′(x).
Check it another way
Expand (3x + 1)² to get 9x² + 6x + 1. Differentiating term by term gives 18x + 6. This is the same as 6(3x + 1).
Expansion is a useful check for this example. The chain rule also works when expanding would be inconvenient.
The easy-to-miss step
If you stop at 2(3x + 1), you have only differentiated the outer function. The inner function changes three times as fast as x, so its derivative contributes the extra factor of 3.
Your turn
Make the idea stick.
Try from memory, then compare your reasoning. Answers stay on this page and aren’t saved or graded.
2 practice questions01
What is the derivative of (2x + 5)³? Identify the inner derivative before simplifying.
Try it in your own words
Compare with the explanation
The outer derivative gives 3(2x + 5)². The inner derivative is 2. Multiply to get 6(2x + 5)².
02
Someone says the derivative of (x² + 1)² is 2(x² + 1). Which step did they miss?
Try it in your own words
Compare with the explanation
They missed multiplying by the inner derivative, 2x. The complete derivative is 2(x² + 1) × 2x = 4x(x² + 1).
Keep for later
Your reference sheet
The key ideas, ready when you need a refresher.
Open the reference sheet
Identify the layers
In f(g(x)), g is the inner function and f is the outer function.
Chain rule
Differentiate the outside, keep the inside, then multiply by the inner derivative.
Formula
The derivative of f(g(x)) is f′(g(x)) × g′(x).
Check the example
The derivative of (3x + 1)² is 2(3x + 1) × 3 = 6(3x + 1).
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