Definition:Difference Quotient

Definition

Let $V$ be a vector space over the real numbers $\R$.

Let $f: \R \to V$ be a function.

A difference quotient is an expression of the form:

$\dfrac {f \left({x + h}\right) - f \left({x}\right)} h$

where $h \ne 0$ is a real number.

Geometric Interpretation

The difference quotient is the slope of the secant line of the graph of $f$ connecting points $P_1 = \left({x, f \left({x}\right)}\right)$ and $P_2 = \left({x+h, f \left({x+h}\right)}\right)$.