Calculus How To

Difference Quotient

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What is a Difference Quotient?

The difference quotient is one way to find a derivative or slope of a function. The formula is:
formal definition

It might look intimidating, but it’s easier to solve than it looks, because many of the terms cancel out.

It’s called a “difference quotient” because the formula has two parts: a difference (subtraction) in the numerator and a quotient (division) between the two parts:
difference quotient


Find the difference quotient for the following function: f(x) = 3x + 2

Step 1: Insert your function into the first part of the formula. In this step, I’m replacing the “f(x+h)” in the left hand part of the numerator with the actual given function, 3x + 2:
difference quotient step 1

Essentially, I just changed the “x” part of 3x + 2 with “x + h”.

Step 2: Replace the “f(x)” in the formula’s numerator. This part is easy, because you’re just copying the formula exactly as written in the question and pasting it in:
step 2 find difference quotient

Step 3: Use algebra to simplify. In most cases, things will start to cancel out, making the formula a little less ugly.

For this particular formula, the first step of simplifying is to remove the parentheses, to get:
step 3 simplifying

From here, you should be able to see that the following terms cancel out:

  • 3x – 3x = 0
  • 2 – 2 = 0

Leaving 3h / h = 3.

The difference quotient (aka the derivative or slope) is 3.

Origins of the Difference Quotient

The formula is derived from the “rise/run” slope formula you’re probably familiar with from algebra:
slope formula

It’s the same formula, with a few substitutions. For example, instead of “y2” in the slope formula, you have “f(x + h)” in the difference quotient formula.

The full list of substitutions:

  • f(x + h) → y2
  • f(x) → y1
  • x → x1
  • x + h → x2
  • h → change in x
  • f(x + h) – f(x) → (y2 – y1)

Substituting those in to the slope formula, you get:
deriving the formula


Difference Quotient. Retrieved October 13, 2019 from:

Stephanie Glen. "Difference Quotient" From Calculus for the rest of us!

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