
Higher order derivatives and higher order differentials 
Higher derivatives of composite functions 
Higher derivatives of composite functions examples 
Higher derivatives of implicit functions 
Higher derivatives of implicit functions examples 






Higher derivatives of composite functions 
If
y = f (u) and
u
= g (x) such that
f
is differentiable at u
and g
is differentiable at x
then,
we differentiate the composite
function y
[u (x)] by
applying the chain rule 
y'
(x)
= y' (u) · u' (x). 
To
get the second derivative of the given composite function we differentiate
the above expression using both the
product rule and the chain rule 
y''
(x)
= y' (u) u'' (x) +
u'
(x) [y'' (u) u' (x)]
= y'' (u)
[u' (x)] ^{2} +
y' (u) u'' (x). 
Proceeding
the same way we get the third derivative of the composite function 
y'''
(x)
= y''' (u)
[u' (x)] ^{3}
+
3y'' (u) u' (x) u'' (x) +
y' (u) u''' (x) and so on. 

Higher derivatives of composite functions
examples 
Example:
Find
the second derivative of 


Solution: 



Example:
Find
the second derivative of y
=
sin^{2}x.

Solution: 
y'
= 2sin x cos x,
y''
= 2[sin x( sin
x)
+ cos
x cos x]
= 2(cos^{2}x

sin^{2}x)
= 2cos 2x. 


Higher derivatives of implicit functions 
The
first derivative of a function y = f(x)
written implicitly as F(x,
y) = 0
is 

In
words,
differentiate the implicit equation with respect to both variables at the
same time such that, when differentiating
with respect to x,
consider
y
constant, while
when differentiating with respect to y,
consider x constant
and multiply by y'. 
We
use this method to differentiate the
above equation to extract the second derivative 

and
after substituting
the value for y' 

Differentiating
the second
derivative equation above, applying the same method, we can get
the third derivative of
an implicit function and so on. 

Higher derivatives of
implicit
functions examples 
Example:
Find
the second derivative of y^{2}
=
2px
or written implicitly
y^{2}

2px
=
0
as
F(x,
y) = 0.

Solution: 
The
first derivative,
2yy'
= 2p
or
yy'
= p,
y'
= p/y. 

The second derivative,
(y' )^{2}
+ yy''
= 0 => y''
= 
(y' )^{2}
/ y 
and
by substituting y'
= p/y,
y''
= 
p^{2}
/ y^{3}.

We
can obtain the same result using the above formulas.
Therefore,
first calculate


and
plug these values into the formulas for the first and the second
derivative 


Example:
Find
the second derivative of y

tan^{1}y
=
x.

Solution:



The second derivative, 



Again,
we
can obtain the same result using the above formulas.











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