Step by step
The question
Question: find dy/dx for y = (5x − 2)².
y = (5x − 2)2
1 The inside
There is something inside: 5x − 2.
Call it u.
u = 5x − 2
Then y is just u².
y = u2
2 The outside
First the outside.
Use the power rule on u².
dydu = 2u^(2 − 1)
We get 2 u¹.
= 2u1
3 The inside's slope
Now the inside.
The power rule on 5x − 2 gives 5.
dudx = 5
4 Multiply
The chain rule: multiply the two.
dy/du times du/dx.
dydx = dydu × dudx
Put the inside back in place of u.
= 2(5x − 2)1 × 5
Careful!
Do not stop here!
We must multiply by the inside's slope, 5.
Careful: not 2(5x − 2)1 ✗
Multiply the numbers in front.
So dy/dx = 10 (5x − 2)¹.
= 10(5x − 2)1
2 × 5 = 10
5 Check
Let's check: the derivative is the slope.
At x = 1, the rule gives 30.
dy/dx at x = 1: 30
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 30.02.
Very close to 30.
Right!
y(1.001) − y(1)0.001 ≈ 30.02
Let's look back.
First: the inside.
Then: the outside.
Then: the inside's slope.
Then: multiply.
Last: check.
Answer
2 × 5 = 10