Step by step
The question
Question: find the derivative of f(x) = 3 x⁴ − 5 x² + 7x − 2.
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.
A number alone gives 0.
Its graph is a flat line, and a flat line has slope 0.
Careful!
A number alone does not stay.
It gives 0.
Work out each one.
So f′(x) = 12 x³ − 10x + 7.
3 Check
Let's check: the derivative is the slope.
At x = 1, the rule gives 9.
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 9.01.
Very close to 9.
Right!
Let's look back.
First: the rule.
Then: each term.
Last: check.