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Applications of the derivative |
Differential
of a function |
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Rules
for differentials |
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Differentials of some
basic functions |
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| Differential
of a function |
| If
given y = f (x)
is differentiable function then, the derivative of y
or f (x) |
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| as
the instantaneous rate of change of y
with respect to x,
can also be written as |
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| where,
the increment
Dx
= dx is called differential of
the independent variable and dy
is the differential of y. |
| Therefore,
the differential of a function |
| dy
= f '(x) dx |
| represent
the main part of the increment Dy
which is linear concerning the increment Dx
= dx, as is shown in
the figure below. |
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| Rules
for differentials |
| Rules
for differentials are the same to those for derivatives, such that |
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1)
dc = 0,
c
is a constant |
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2)
dx = Dx,
x
is the independent variable |
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3)
d (cu) = c
du |
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4)
d (u ±
v) = du ±
dv |
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5)
d (u v) = u
dv ±
v du |
| 6) |
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7) d
f (u) = f ' (u) du. |
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Differentials
of some basic functions |
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1) d
un = n
un -
1du |
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2) d
eu = eu
du |
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3)
d
an = an ln a
du |
| 4) |
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5) d
sin u = cos u
du |
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6) d
cos u = -
sin u
du |
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| 8) |
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| 9) |
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| 11) |
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| 12) |
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Example:
To
show the use of the formula 2) d
eu = eu du
above, let substitute;
a) u = x,
b) u = cos
x
and
c) u = x3.
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Solution:
a)
By substituting u = x,
obtained is d
ex = ex dx
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b) By substituting u =
cos x,
we get d
ecos x = ecos x
d(cos x) =
-
ecos x sin x
dx |
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c)
By substituting
u = x3,
we get |
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| Calculus contents
D |
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| Copyright
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