I
NTRODUCTION
TO
D
ERIVATIVES
Tangent Slope as a Function/Differentiation
So what happens if you are not given the specific point at which we want to find the tangent?
We can find a function to define the tangent at any point on the given curve.
Let’s try finding the tangent slope at some arbitrary point.
Ex
.
2
)
(
x
x
f
=
at the point
)
,
(
2
x
x
P
(ie. an arbitrary point)
A different point Q on the graph
has coordinates
(
29
2
)
(
,
h
x
h
x
+
+
.
0
≠
h
x
x
h
h
x
h
h
x
h
xh
x
h
x
h
x
slopeofPQ
h
h
h
h
2
)
0
(
2
)
2
(
lim
2
lim
)
(
)
(
lim
)
(
lim
0
2
2
2
0
2
2
0
0
=
+
=
+
=

+
+
=

+
=
→
→
→
→
Therefore the slope of the function
2
)
(
x
x
f
=
at the point
)
,
(
2
x
x
P
is
x
2
, another function.
This is because the slope of the function changes as the value of
x
changes.
We call this
function the
derivative
of
)
(
x
f
.
We denote this
)
(
x
f
′
“
f
prime
x
”.
In the example, the
derivative of
2
)
(
x
x
f
=
is
x
x
f
2
)
(
=
′
.
The process of calculating the derivative of a function is called
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 Fall '08
 Kim
 Calculus, Derivative, Slope, lim

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