# Difference Quotient

This is the "Difference Quotient":

\frac{f(x+Δx) − f(x)}{Δx}

It gives the **average** slope between two points on a curve **f(x)** that are **Δx** apart:

### Example: find the average slope of **f(x) = x**^{2} − 2x + 1

at **x = 3** and **Δx = 0.1**

^{2}− 2x + 1

Evaluate f(x) at x=3:

f(3) = 3^{2 }− 2×3 + 1 = **4**

Now for f(x+Δx):

f(3.1) = (3.1)^{2 }− 2×3.1 + 1 = **4.41**

And the Difference Quotient is:

\frac{f(3.1) − f(3)}{0.1} = \frac{4.41 − 4}{0.1} = \frac{0.41}{0.1} = **4.1**

Let's try a smaller value of Δx:

### Example continued: try Δx = 0.01

f(3.01) = (3.01)^{2 }− 2×3.01 + 1 = **4.0401**

So we have:

\frac{f(3.01) − f(3)}{0.01} = \frac{4.0401 − 4}{0.01} = \frac{0.0401}{0.01} = **4.01**

As Δx heads towards 0, the value of the slope heads towards the true slope at that point.

In this case, as Δx gets smaller the slope **seems to be** heading towards **4**, right?

Well, that is the idea behind derivatives, which can find the answer exactly (without guesses) by having Δx head towards 0.