Step by step
The question
Question: find the derivative of f(x) = 4 x³ − 5 x² + 7.
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: 4 x³.
Its power is 3.
The power comes down in front: 4 times 3.
And the new power is 3 − 1.
3 − 1 is 2.
4 times 3 is 12.
So 4 x³ gives 12 x².
Next: −5 x².
Its power is 2.
The power comes down in front: 5 times 2.
And the new power is 2 − 1.
2 − 1 is 1.
5 times 2 is 10.
So −5 x² gives −10x.
Last: 7.
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) = 12 x² − 10x.
3 Check
Let's check: the derivative is the slope.
At x = 1, the rule gives 2.
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 2.01.
Very close to 2.
Right!
4 Pick the choice
Our answer is f′(x) = 12x² − 10x.
That is choice A.
5 Why not the others
B is 12x² − 10x + 7.
Wrong: kept the 7: the derivative of a number is 0.
C is 4x² − 5x.
Wrong: did not bring the powers down.
D is 12x³ − 10x².
Wrong: brought the powers down but did not take one away.
Let's look back.
First: the rule.
Then: each term.
Then: check.
Then: pick the choice.
Last: why not the others.