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The
equation of the line tangent to the given curve at the given point
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The
derivative of a function f
(x)
is a new function that contains the value of the derivative of
all points on the original
function.
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| That is, the
derivative function gives the slope of
the line tangent to f
(x)
at every point x. |
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The
slope of the tangent line provides information about how the
graph of the function is changing.
|
| Thus,
if
f
'
(x)
<
0
then f
(x)
is decreasing |
|
and if
f
'
(x)
>
0 then
f
(x)
is increasing, as is shown in the
figure
below. |
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The
process of determining or finding the derivative is called differentiation.
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In
order a function to be differentiable, it must be continuous,
and both one-sided limits (the
left-handed and right-handed limits) must
be equal at
the given point.
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For example, the function f
(x)
= | x | is not differentiable
at x
= 0, although
it is continuous there.
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Derivatives
of basic or elementary functions
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Determining
the derivative of a function as the limit of the difference
quotient
|
| We
use the limit definition |
 |
|
| to
find the derivative of a function.
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The
derivative of the power function
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The
derivative of the linear function
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Thus,
for n = 1,
that is for the linear function f
(x)
= x, the
difference
f
(x
+ h) -
f
(x) = x
+ h - x
= h so that the
difference quotient equals 1.
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Therefore,
if f
(x)
= x then |
 |
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The
derivative of a constant
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The
function f
(x)
= c,
where c
is a fixed constant, is graphically represented by a horizontal
line so that at any
given point (x,
f
(x)) the slope
of the line tangent to the graph of f
is 0. Therefore,
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| f
(x)
= c and
f
(x
+ h)
= c
so
that f
(x
+ h) -
f
(x)
= c - c
= 0. |
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Thus, |
 |
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that
is, if f
(x)
= c
t
hen |
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