The Difference Quotient

What Problem Are We Solving?

Imagine you’re driving. Your speedometer shows your speed right now.

But what if you only had the odometer (total distance)?

You could calculate your average speed:

$\text{Average speed} = \frac{\text{change in distance}}{\text{change in time}}$

The shorter the time interval, the closer this gets to your actual speed.

The Same Idea for Functions

For any function $f(x)$, we can measure how fast it’s changing.

The difference quotient gives the average rate of change between two points:

$\frac{f(x + h) - f(x)}{h}$

Breaking it down:

• $f(x)$ — output at point $x$
• $f(x + h)$ — output at a nearby point, $h$ units to the right
• $f(x + h) - f(x)$ — how much the output changed
• Divide by $h$ — change per unit of input

In short: change in output ÷ change in input

Example: $f(x) = x^2$

Step 1: Find $f(x + h)$

Replace every $x$ with $(x + h)$:

Step 2: Subtract $f(x)$

Step 3: Divide by $h$

Result: The difference quotient for $f(x) = x^2$ is $2x + h$.

This tells us the average rate of change between $x$ and $x + h$.

The Connection to Calculus

What happens as $h$ gets smaller?

• When $h = 1$: difference quotient $= 2x + 1$
• When $h = 0.1$: difference quotient $= 2x + 0.1$
• When $h = 0.01$: difference quotient $= 2x + 0.01$

As $h \to 0$, the difference quotient approaches $2x$.

This limit is called the derivative. It measures the instantaneous rate of change.

Simpler Example: $f(x) = 3x + 1$

Step 1: $f(x + h) = 3(x + h) + 1 = 3x + 3h + 1$

Step 2:

Step 3: $\dfrac{3h}{h} = 3$

Result: The difference quotient is just $3$.

This makes sense — a line has a constant slope, so the rate of change is the same everywhere.

Why This Matters

The difference quotient is the foundation of calculus.

• Now: Practice the algebra of computing it
• Later: Take the limit as $h \to 0$ to find derivatives

Master this, and derivatives become straightforward.