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