Step by step
The question
Question: find dy/dx for y = (x² − 5)³.
y = (x2 − 5)3
1 The inside
There is something inside: x² − 5.
Call it u.
u = x2 − 5
Then y is just u³.
y = u3
2 The outside
First the outside.
Use the power rule on u³.
dydu = 3u^(3 − 1)
We get 3 u².
= 3u2
3 The inside's slope
Now the inside.
The power rule on x² − 5 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.
= 3(x2 − 5)2 × 2x
Careful!
Do not stop here!
We must multiply by the inside's slope, 2x.
Careful: not 3(x2 − 5)2 ✗
Multiply the numbers in front.
So dy/dx = 6x (x² − 5)².
= 6x(x2 − 5)2
3 × 2x = 6x
5 Check
Let's check: the derivative is the slope.
At x = undefined, the rule gives NaN.
dy/dx at x = undefined: NaN
Now the slope between two very close points: x = undefined and x = NaN.
A calculator gives about NaN.
Very close to NaN.
Right!
y(NaN) − y(undefined)0.001 ≈ NaN
Let's look back.
First: the inside.
Then: the outside.
Then: the inside's slope.
Then: multiply.
Last: check.
Answer
3 × 2x = 6x