Step by step
The question
Question: find dy/dx for y = (x² + 2x)³.
y = (x2 + 2x)3
1 The inside
There is something inside: x² + 2x.
Call it u.
u = x2 + 2x
Then y is just u³.
y = u3
2 The outside
First the outside.
Use the power rule on u³.
dydu = 3u^(3 − 1)
We get 3 u².
= 3u2
3 The inside's slope
Now the inside.
The power rule on x² + 2x gives 2x + 2.
dudx = 2x + 2
4 Multiply
The chain rule: multiply the two.
dy/du times du/dx.
dydx = dydu × dudx
Put the inside back in place of u.
= 3(x2 + 2x)2 × (2x + 2)
Careful!
Do not stop here!
We must multiply by the inside's slope, 2x + 2.
Careful: not 3(x2 + 2x)2 ✗
So dy/dx = 3(2x + 2) (x² + 2x)².
= 3(2x + 2)(x2 + 2x)2
5 Check
Let's check: the derivative is the slope.
At x = 1, the rule gives 108.
dy/dx at x = 1: 108
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 108.17.
Very close to 108.
Right!
y(1.001) − y(1)0.001 ≈ 108.17
Let's look back.
First: the inside.
Then: the outside.
Then: the inside's slope.
Then: multiply.
Last: check.
Answer
= 3(2x + 2)(x2 + 2x)2