Step by step
The question
Question: find dy/dx for y = (x² + 1)⁴.
y = (x2 + 1)4
1 The inside
There is something inside: x² + 1.
Call it u.
u = x2 + 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 x² + 1 gives 2x.
dudx = 2x
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(x2 + 1)3 × 2x
Careful!
Do not stop here!
We must multiply by the inside's slope, 2x.
Careful: not 4(x2 + 1)3 ✗
Multiply the numbers in front.
So dy/dx = 8x (x² + 1)³.
= 8x(x2 + 1)3
4 × 2x = 8x
5 Check
Let's check: the derivative is the slope.
At x = 0, the rule gives 0.
dy/dx at x = 0: 0
Now the slope between two very close points: x = 0 and x = 0.001.
A calculator gives about 0.
Very close to 0.
Right!
y(0.001) − y(0)0.001 ≈ 0
Let's look back.
First: the inside.
Then: the outside.
Then: the inside's slope.
Then: multiply.
Last: check.
Answer
4 × 2x = 8x