Step by step
The question
Question: find dy/dx for y = (3x + 1)⁴.
y = (3x + 1)4
1 The inside
There is something inside: 3x + 1.
Call it u.
u = 3x + 1
Then y is just u⁴.
y = u4
2 The outside
First the outside.
Use the power rule on u⁴.
dydu = 4u^(4 − 1)
We get 4 u³.
= 4u3
3 The inside's slope
Now the inside.
The power rule on 3x + 1 gives 3.
dudx = 3
4 Multiply
The chain rule: multiply the two.
dy/du times du/dx.
dydx = dydu × dudx
Put the inside back in place of u.
= 4(3x + 1)3 × 3
Careful!
Do not stop here!
We must multiply by the inside's slope, 3.
Careful: not 4(3x + 1)3 ✗
Multiply the numbers in front.
So dy/dx = 12 (3x + 1)³.
= 12(3x + 1)3
4 × 3 = 12
5 Check
Let's check: the derivative is the slope.
At x = 0, the rule gives 12.
dy/dx at x = 0: 12
Now the slope between two very close points: x = 0 and x = 0.001.
A calculator gives about 12.05.
Very close to 12.
Right!
y(0.001) − y(0)0.001 ≈ 12.05
Let's look back.
First: the inside.
Then: the outside.
Then: the inside's slope.
Then: multiply.
Last: check.
Answer
4 × 3 = 12