Step by step
The question
Question: find the derivative of f(x) = 6 x² − 4x + 9.
There are four choices.
First we solve it, the normal way.
Then we pick.
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: 6 x².
Its power is 2.
The power comes down in front: 6 times 2.
And the new power is 2 − 1.
2 − 1 is 1.
6 times 2 is 12.
So 6 x² gives 12x.
Next: −4x.
x alone gives 1: its graph goes up 1 for each 1 across, so its slope is 1.
4 times 1 is 4.
Last: 9.
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) = 12x − 4.
3 Check
Let's check: the derivative is the slope.
At x = 1, the rule gives 8.
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 8.01.
Very close to 8.
Right!
4 Pick the choice
Our answer is f′(x) = 12x − 4.
That is choice C.
5 Why not the others
A is 6x − 4.
Wrong: did not bring the power 2 down.
B is 12x + 5.
Wrong: kept the 9: the derivative of a number is 0.
D is 2x³ − 2x² + 9x.
Wrong: integrated instead.
Let's look back.
First: the rule.
Then: each term.
Then: check.
Then: pick the choice.
Last: why not the others.