Step by step
The question
Question: find the derivative of f(x) = 6 x^(2/3).
1 The rule
From the definition we found: x² gives 2x, and x³ gives 3x².
See the pattern?
The power comes down in front, and the power goes down by 1.
This is the power rule.
We write the derivative as f′(x), "f prime of x".
2 Each term
Use the rule on each term.
A number in front just waits, and multiplies.
Take 1 away from the power: 2/3 − 1 = −1/3.
Careful!
The power must go down by 1 too.
Work out each one.
3 Tidy up
Books write it without minus or fraction powers.
So f′(x) = 4 over (x^(1/3)).
4 Check
Let's check: the derivative is the slope.
At x = 1, the rule gives 4.
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 4.
Very close to 4.
Right!
Let's look back.
First: the rule.
Then: each term.
Then: tidy up.
Last: check.