Step by step
The question
Question: find the derivative of f(x) = 3 over x.
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 Write as powers
To use the rule, every x needs a power.
x to a power at the bottom is the same as a minus power on top.
3 Each term
The term: 3 x⁻¹.
Its power is −1.
The power comes down in front: 3 times −1.
And the new power is −1 − 1.
−1 − 1 is −2.
3 times −1 is 3.
So 3 x⁻¹ gives −3 x⁻².
Careful!
Taking 1 away from −1 makes it −2: the power goes down, never up.
So: f′(x) = −3 x⁻².
4 Tidy up
The usual way to write the answer: no minus or fraction powers.
So f′(x) = −3/x².
5 Check
Let's check: the derivative is the slope.
At x = 1, the rule gives −3.
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about −3.
Very close to −3.
Right!
6 Pick the choice
Our answer is f′(x) = −3/x².
That is choice A.
7 Why not the others
B is −3/x.
Wrong: the power did not go down.
C is 3ln x.
Wrong: that is the integral.
D is 3/x².
Wrong: lost the minus: x⁻¹ gives −x⁻².
Let's look back.
First: the rule.
Then: write as powers.
Then: each term.
Then: tidy up.
Then: check.
Then: pick the choice.
Last: why not the others.