difference quotient

The expression $\dfrac{f(x+h)-f(x)}{h}$ — the slope of a secant line through two nearby points on a curve.

The difference quotient of $f$ at $x$ is $\dfrac{f(x+h)-f(x)}{h}$ for $h \neq 0$ — the slope of the secant line joining $(x, f(x))$ and $(x+h, f(x+h))$ . Letting $h \to 0$ turns this algebraic quotient into the derivative.

Examples

For $f(x)=x^2$ , the difference quotient is $\dfrac{(x+h)^2-x^2}{h} = 2x+h$ .