difference quotient

English

Noun

difference quotient (plural difference quotients)

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

Translations

difference quotient

• Asturian: cociente diferencial m
• Catalan: quocient de diferències m
• Chinese:

  Mandarin: 差商 (zh) (chāshāng)

• Danish: differenskvotient c
• Dutch: differentiaalquotiënt (nl) n
• Finnish: erotusosamäärä (fi)
• French: quotient des différences m, quotient des variations m
• Galician: cociente da diferenza m
• German: Differenzenquotient m
• Italian: rapporto incrementale (it) m
• Portuguese: quociente de diferenças m
• Spanish: cociente incremental m, cociente de diferencias m

See also

• derivative
• difference equation