Step by step
The question
Question: find dy/dx for y = x³ − 6x.
y = x3 − 6x
1 The rule
From the definition we found: x² gives 2x, and x³ gives 3x².
x2 → 2x
x3 → 3x2
See the pattern?
The power comes down in front, and the power goes down by 1.
This is the power rule.
ddxxn = nxn − 1
dy/dx is another way to write the derivative of y.
Say it "d y by d x".
dy/dx: the derivative of y
2 Each term
Use the rule on each term.
A number in front just waits, and multiplies.
dydx = 3x^(3 − 1) − 6 × 1
Careful!
The power must go down by 1 too.
Careful: not x3 → 3x3 ✗
Work out each one.
So dy/dx = 3 x² − 6.
= 3x2 − 6
3 Check
Let's check: the derivative is the slope.
At x = 1, the rule gives −3.
dy/dx at x = 1: 3 − 6 = −3
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about −3.
Very close to −3.
Right!
y(1.001) − y(1)0.001 ≈ −3
Let's look back.
First: the rule.
Then: each term.
Last: check.
Answer
= 3x2 − 6