First year:
Differentiation: chain rule
So far we have been differentiating simple functions, like x 3 , e x or cos(x) but
most of the functions you are likely to encounter will be composite functions
3
like e x , cos(e x ) etc… The chain rule tells you how to differentiate such
functions.
(I)
The general chain rule
d
 f g ( x)   d g ( x)   df g ( x) .
dx
dx
 dx

Another notation for the derivative of a function f (x) is f ' ( x) . With this
notation, the chain rule can be written as follows:
 f g ( x)'  g ' ( x)  f ' ( g ( x)).
Let us consider some examples:
-
h( x)  e x
3
In this case, we take f ( x)  e x and g ( x)  x 3 . f g (x) means here that you
take the exponential of x 3 .
We have
d
d
f ( x)  e x and
g ( x)  3x 2 .
dx
dx
Let’s apply the chain rule, now:
3
d
d
h( x)   f g ( x)   3x 2 e x .
dx
dx
-
h( x)  cos( x 3 ) .
Here f ( x)  cos( x) and g ( x)  x 3 .
d
d
f ( x)   sin( x) and
g ( x)  3 x 2 , so that
We have
dx
dx
d
h( x)  3x 2 sin( x 3 ).
dx
-
h( x)  cos 3 ( x).
In this case f ( x)  x 3 and g ( x)  cos( x) (because h is the cube of the cos ).
With
-
d
d
f ( x)  3 x 2 and
g ( x)   sin( x) , the chain rule yields
dx
dx
d
2
h( x)  3 sin( x)cos( x)  .
dx
h( x)  log( 1  x 2 )
We take f ( x)  log( x) and g ( x)  1  x 2 , so that
Hence
-
d
1
d
f ( x )  and
g ( x)  2 x .
dx
x
dx
d
2x
h( x ) 
.
dx
1 x2
h( x)  1  cos( x)
Here f ( x)  x and g ( x)  1  cos( x) , so that
d
d
1
g ( x)   sin( x) .
and
f ( x) 
dx
dx
2 x
The chain rule gives:
d
 sin( x)
.
h( x ) 
dx
2 1  cos( x)
The chain rule require quite a bit of practice to be handled properly, so here
are a few exercises:
Differentiate the following functions:
-
h( x)  e 2 x
-
h( x)  sin( 1  x 2 )
-
h( x)  log( 2  cos( x)  sin( x))
-
h( x)  e x  x  1
2
Sometimes, you will even be required to use various rules at the same time
in order to differentiate a function. See if you can differentiate this one:
-
h( x ) 
x2
.
cos 2 ( x)  1


Power chain rule
(II)
The power chain rule is really a special case of the general chain rule, but it is
comparatively easier to use.
The power chain rule tells you how to differentiate functions of the form
n
h( x)  g ( x)  . Taking f ( x)  x n in the formula for the general chain rule, we
obtain
d
d

n 1
h( x)  n g ( x)   g ( x)  .
dx
 dx

h( x )   g ( x )   f  g ( x )  
n
Power chain rule:
h( x )   g ( x ) 
n

d
d

n 1
h( x)  n g ( x)   g ( x)  .
dx
 dx

or, with the alternative notation for the derivative
h( x )   g ( x ) 
n
 h' ( x)  ng ' ( x)g ( x)  .
n 1
Let’s see how it works in practice:
-
h( x)  cos10 ( x) .
Here g ( x)  cos( x) , so that
-

d
d
g ( x)   sin( x) and
h( x)  10 sin( x) cos 9 ( x) .
dx
dx

5
h( x )  3 x 2  2 x  1
If you set g ( x)  3x 2  2 x  1, then
yields

d
g ( x)  6 x  2 and the power chain rule
dx

4
d
h( x)  56 x  2  3 x 2  2 x  1 .
dx
Finally, a few exercises to practice.
Differentiate the following functions:
-
h( x)  3sin 4 ( x).

h( x)  x
h( x)  4 x 5  3x 3  6
2

7

3
 cos( x)  sin 2 ( x) 
1
.
( x  2) 2