Step by step
The question
Question: find the derivative of f(x) = x⁵.
f(x) = x5
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
We write the derivative as f′(x), "f prime of x".
f′(x): the derivative of f(x)
2 Each term
Use the rule on each term.
f′(x) = 5x^(5 − 1)
Careful!
The power must go down by 1 too.
Careful: not x5 → 5x5 ✗
Work out each one.
So f′(x) = 5 x⁴.
= 5x4
3 Check
Let's check: the derivative is the slope.
At x = 1, the rule gives 5.
f′(1) = 5
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 5.01.
Very close to 5.
Right!
f(1.001) − f(1)0.001 ≈ 5.01
Let's look back.
First: the rule.
Then: each term.
Last: check.
Answer
= 5x4