Step by step
The question
Question: find dy/dx for y = 2 (3x − 1)⁴.
y = 2(3x − 1)4
1 The inside
There is something inside: 3x − 1.
Call it u.
u = 3x − 1
Then y is just 2 u⁴.
y = 2u4
2 The outside
First the outside.
Use the power rule on 2 u⁴.
dydu = 2 × 4u^(4 − 1)
We get 8 u³.
= 8u3
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.
= 8(3x − 1)3 × 3
Careful!
Do not stop here!
We must multiply by the inside's slope, 3.
Careful: not 8(3x − 1)3 ✗
Multiply the numbers in front.
So dy/dx = 24 (3x − 1)³.
= 24(3x − 1)3
8 × 3 = 24
5 Check
Let's check: the derivative is the slope.
At x = 1, the rule gives 192.
dy/dx at x = 1: 192
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 192.43.
Very close to 192.
Right!
y(1.001) − y(1)0.001 ≈ 192.43
Let's look back.
First: the inside.
Then: the outside.
Then: the inside's slope.
Then: multiply.
Last: check.
Answer
8 × 3 = 24