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The derivative of a function as the limit of the difference
quotient
|
| We
use the limit definition |
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|
| to
find the derivative of a function.
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The
derivative of the sine function
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|
We
use the limit of the
difference quotient to find the
derivative of the function f
(x)
= sin x. |
| Let
rewrite the
difference quotient applying the sum to product formula,
|
 |
| Since,
the derivative is the limit of the difference quotient as h
tends to zero then, |
 |
| Therefore,
if f
(x)
= sin x then |
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|
The
derivative of the cosine function
|
| We
use the limit of the
difference quotient to find the
derivative of the function f
(x)
= cos x. |
| Let
rewrite the
difference quotient applying the sum to product formula,
|
 |
| Since,
the derivative is the limit of the difference quotient as h
tends to zero then, |
 |
| Therefore,
if f
(x)
= cos x
then |
 |
|
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|
The
derivative of the exponential function
|
| Let
use the limit of the
difference quotient to find the
derivative of the function f
(x)
= ax. |
| Since
the
difference quotient is |
 |
|
| then,
the derivative as the limit of the difference quotient as h
tends to zero |
 |
|
That
is, by plugging t
= ah -
1, then t
®
0
as h ®
0,
and
|
 |
 |
| Therefore,
if f
(x)
= ax then |
 |
|
| or
when the base a
is substituted by the natural base e
obtained is the exponential function ex,
thus |
|
if f
(x)
= ex then |
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The
derivative of the logarithmic function
|
| Let
use the limit of the
difference quotient to find the
derivative of the function f
(x)
= loga x. |
| The
difference quotient applied to the given function |
 |
| As
the derivative is the limit of the difference quotient as h
tends to zero, then
|
 |
| Then, applying the base
change identity and substituting a
= e
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 |
|
if f
(x)
= loga x then |
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|
if
f
(x)
= ln x then |
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