Step by step
The question
Question: find the derivative of f(x) = x³ − 3 x² − 9x + 5.
1 The rule
d/dx means: the derivative of.
Here, of x to the power n.
The rule: the power n comes down in front, and the new power is n − 1.
We write the derivative as f′(x), "f prime of x".
2 Each term
First: x³.
Its power is 3.
The power comes down in front.
And the new power is 3 − 1.
3 − 1 is 2.
So x³ gives 3 x².
Next: −3 x².
Its power is 2.
The power comes down in front: 3 times 2.
And the new power is 2 − 1.
2 − 1 is 1.
3 times 2 is 6.
So −3 x² gives −6x.
Next: −9x.
x alone gives 1: its graph goes up 1 for each 1 across, so its slope is 1.
9 times 1 is 9.
Last: 5.
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.
We put the terms together: f′(x) = 3 x² − 6x − 9.
3 Check
Let's check: the derivative is the slope.
At x = 1, the rule gives −12.
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about −12.
Very close to −12.
Right!
Let's look back.
First: the rule.
Then: each term.
Last: check.