Step by step
The question
Question: find dy/dx for y = (4x + 3)³.
y = (4x + 3)3
1 The inside
There is something inside: 4x + 3.
Call it u.
u = 4x + 3
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 4x + 3 gives 4.
dudx = 4
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(4x + 3)2 × 4
Careful!
Do not stop here!
We must multiply by the inside's slope, 4.
Careful: not 3(4x + 3)2 ✗
Multiply the numbers in front.
So dy/dx = 12 (4x + 3)².
= 12(4x + 3)2
3 × 4 = 12
5 Check
Let's check: the derivative is the slope.
At x = 0, the rule gives 108.
dy/dx at x = 0: 108
Now the slope between two very close points: x = 0 and x = 0.001.
A calculator gives about 108.14.
Very close to 108.
Right!
y(0.001) − y(0)0.001 ≈ 108.14
Let's look back.
First: the inside.
Then: the outside.
Then: the inside's slope.
Then: multiply.
Last: check.
Answer
3 × 4 = 12