Step by step
The question
Question: find the derivative of f(x) = 2 x³ + 6x − 1.
f(x) = 2x3 + 6x − 1
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.
A number in front just waits, and multiplies.
f′(x) = 2 × 3x^(3 − 1) + 6 × 1 + 0
A number alone gives 0.
Its graph is a flat line, and a flat line has slope 0.
a number alone → 0
Careful!
A number alone does not stay.
It gives 0.
Careful: not −1 → −1 ✗
Work out each one.
So f′(x) = 6 x² + 6.
= 6x2 + 6
3 Check
Let's check: the derivative is the slope.
At x = 1, the rule gives 12.
f′(1) = 6 + 6 = 12
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 12.01.
Very close to 12.
Right!
f(1.001) − f(1)0.001 ≈ 12.01
Let's look back.
First: the rule.
Then: each term.
Last: check.
Answer
= 6x2 + 6