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Differential
calculus - derivatives
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Higher derivatives of parametric functions
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Higher derivatives of parametric functions
example
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Example:
Find
the second derivative of the parametric functions x
= ln t and
y
= t3
+ 1.
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Solution:
Since |
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then |
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Higher order differentials
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The
first order differential of a function y
= f (x)
at a point x |
dy
= f '(x)dx = y'dx |
is
defined as the linear part of the increment of the function expressed as
the product of the derivative of the function
and the corresponding increment of the independent variable. |
The
second order differential is defined as the differential of the first
order differential with respect to |
the same increment dx,
written d
2 y
= d(dy) = d(y'dx) = y'' (dx)2. |
Similarly defined
are the third and higher order differentials, d
3 y
= y''' (dx)3, |
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·
·
·
·
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therefore,
d
n y
= y (n) (dx)n. |
Furthermore,
if y
= f (u), where
u = g (x),
then |
d
2 y
= y'' (du)2 +
y' d
2 u, |
d
3 y
= y''' (du)3 +
3 y''
du d 2 u
+
y' d
3 u,
etc. |
Note
that, here primes denote derivatives with respect to u. |
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Higher order differentials examples
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Example:
Find
d
2 y
of the function y
= cos 3x.
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Solution:
Therefore, given
y
= f(u)
= cos
u,
where
u
= g(x)
= 3x.
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The
second differential we calculate using the formula |
d
2 y
= y'' (du)2 +
y' d
2 u,
where d
2 u
= g'' (x) (dx)2. |
Since,
y' (u)
=
-
3sin
3x
then, y''
(u)
=
-
9cos
3x |
and
g' (x)
= (3x)' = 3
then, g''
(x) = 0,
so that d
2 u
= 0. Thus,
d
2 y
= -
9cos
3x
(du)2. |
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Example:
Find
d
2 y
of the
function y
= sin x
· ln x.
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Solution:
The second differential
d
2 y
= y'' (dx)2,
since
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Contents
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