Step by step
The question
Question: find f′(x) for f(x) = x² + 5x, from the definition.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 The idea
Look at the curve y = x² + 5x.
Take a point P on it.
The derivative tells how steep it is at P.
Take a second point Q, a step h to the right.
The line through P and Q has slope rise over run.
When h goes to 0, Q slides onto P.
The line just touches the curve: the tangent.
Its slope is the derivative.
The definition: the rise is f(x + h) minus f(x).
The run is h.
Then h goes to 0.
2 Find f(x + h)
Put x + h in place of x, in brackets.
(x + h)² means (x + h) times (x + h).
Open it: x² + 2xh + h².
Careful!
The 2xh in the middle must not be lost.
What is in front of a bracket multiplies every part inside.
Now we have x² + 2xh + h² + 5x + 5h.
3 Take away f(x)
Now take away f(x).
It goes in brackets: − (x² + 5x).
A minus in front of a bracket changes the sign of every part inside.
Every term of f(x) meets its opposite.
They make zero: cross them out.
We write it again, without the crossed-out parts.
Every term left has an h.
4 Divide by h
Divide by the run, h.
Every term on top has an h, so take it out: 2xh + h² + 5h is h(2x + h + 5).
h is close to 0, but not 0.
So h/h is 1.
Cross them out.
We write it again, without the crossed-out parts.
5 Let h go to 0
Now let h go to 0.
Every term with an h becomes 0.
Cross them out.
So the derivative is f′(x) = 2x + 5.
6 Check
Let's check on the picture.
P is at x = 1.
The rule gives f′(1) = 7.
From P, go right 1.
Then up 7.
We land on the tangent.
Right!
7 Pick the choice
Our answer is f′(x) = 2x + 5.
That is choice B.
8 Why not the others
A is 2x.
Wrong: lost the + 5x.
C is x + 5.
Wrong: the derivative of x² is 2x, not x.
D is 0.
Wrong: put in h = 0 before simplifying.
Let's look back.
First: the idea.
Then: find f(x + h).
Then: take away f(x).
Then: divide by h.
Then: let h go to 0.
Then: check.
Then: pick the choice.
Last: why not the others.