Difference quotient (and why you cannot divide by 0)

The difference quotient is the average rate of change from $x$ to $x+h$.

Setting $h=0$ directly gives $\frac{0}{0}$, which is undefined; that is the wall. So keep $h\neq 0$, expand the numerator, cancel $h$ to remove the wall, and then let $h$ approach 0.

For $f(x)=x^2$ the numerator is $2xh+h^2$, cancelling gives $2x+h$, and as $h\to 0$ this becomes $2x$. This procedure is exactly the definition of the derivative.

Examples

1. 01Simplify (h≠0):$\displaystyle \frac{(x+h)^2-x^2}{h}$AnswerHide

   Answer$\displaystyle 2x+h$
2. 02Difference quotient at $x=3$. Divide by $h$:$\displaystyle \frac{6h+h^2}{h}$AnswerHide

   Answer$\displaystyle 6+h$
3. 03With $f(x)=x^2$, cancel $h$:$\displaystyle \frac{f(5+h)-f(5)}{h}$AnswerHide

   Answer$\displaystyle 10+h$
4. 04With $f(x)=3$, simplify (h≠0):$\displaystyle \frac{f(x+h)-f(x)}{h}$AnswerHide

   Answer$\displaystyle 0$
5. 05$f(x)=x^2$. Expand and simplify:$\displaystyle f(3+h)-f(3)$AnswerHide

   Answer$\displaystyle 6h+h^2$
6. 06With $f(x)=x^2$, cancel $h$:$\displaystyle \frac{f(3+h)-f(3)}{h}$AnswerHide

   Answer$\displaystyle 6+h$

FAQ

Q1Why can't I substitute h=0 right away?Show answerHide

A

The denominator becomes $0$, and $\frac{0}{0}$ has no defined value. Cancel $h$ while $h\neq 0$ so it leaves the denominator; then the value as $h\to 0$ can be read off.

Practice this pattern in the app, 5 problems a day.

Published 2026-09-04 · Updated 2026-09-05