Step by step
The question
Question: find dy/dx for y = (x + 4)⁵.
y = (x + 4)5
1 The inside
There is something inside: x + 4.
Call it u.
u = x + 4
Then y is just u⁵.
y = u5
2 The outside
First the outside.
Use the power rule on u⁵.
dydu = 5u^(5 − 1)
We get 5 u⁴.
= 5u4
3 The inside's slope
Now the inside.
The power rule on x + 4 gives 1.
dudx = 1
4 Multiply
The chain rule: multiply the two.
dy/du times du/dx.
dydx = dydu × dudx
Put the inside back in place of u.
= 5(x + 4)4 × 1
Careful!
Do not stop here!
We must multiply by the inside's slope, 1.
Careful: not 5(x + 4)4 ✗
Multiply the numbers in front.
So dy/dx = 5 (x + 4)⁴.
= 5(x + 4)4
5 × 1 = 5
5 Check
Let's check: the derivative is the slope.
At x = 0, the rule gives 1280.
dy/dx at x = 0: 1280
Now the slope between two very close points: x = 0 and x = 0.001.
A calculator gives about 1280.64.
Very close to 1280.
Right!
y(0.001) − y(0)0.001 ≈ 1280.64
Let's look back.
First: the inside.
Then: the outside.
Then: the inside's slope.
Then: multiply.
Last: check.
Answer
5 × 1 = 5