Difference Quotient

This is the "Difference Quotient":

f(x+Δx) − f(x)Δx

It gives the average slope between two points on a curve f(x) that are Δx apart, and is used with derivatives:


slope dy/dx


Example: f(x) = x2 − 2x + 1 at x = 3 and Δx = 0.1

Evaluate f(x) at x=3:

f(3) = 32 − 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:

f(3.1) − f(3)0.1 = 4.41 − 40.1 = 0.410.1 = 4.1

 As Δx heads towards 0, the value of the slope heads towards the true slope at x

Example continued: try Δx = 0.01


f(3.01) = (3.01)2 − 2×3.01 + 1 = 4.0401

So we have:

f(3.01) − f(3)0.01 = 4.0401 − 40.01 = 0.04010.01 = 4.01

As Δx gets smaller the slope seems to be heading towards 4, right? That is the idea behind derivatives (which can find the answer exactly).