difference quotient

difference quotient (plural difference quotients)

1. (mathematical analysis) A mathematical quantity of the form ${\displaystyle {\frac {f(a+h)-f(a)}{h}}}$ for some function ${\displaystyle f(x)}$, having two main interpretations:
   1. (calculus) The average rate of change of the function ${\displaystyle f(x)}$ over the interval ${\displaystyle [a,\,a+h]}$. As ${\displaystyle h}$ approaches zero, the value of ${\displaystyle {\frac {f(a+h)-f(a)}{h}}}$ approaches the derivative or instantaneous rate of change of ${\displaystyle f(x)}$ at ${\displaystyle x=a}$.
   2. (geometry) The slope of the secant line passing through the points ${\displaystyle \left(a,\,f(a)\right)}$ and ${\displaystyle \left(a+h,\,f(a+h)\right)}$. As the horizontal distance ${\displaystyle h>0}$ between the points approaches zero, the slope of this secant line approaches the slope of the tangent line to ${\displaystyle f(x)}$ at ${\displaystyle x=a}$, which is equal to the derivative of ${\displaystyle f(x)}$ at ${\displaystyle x=a}$.

• derivative
• difference equation