1
Differentiation 10
chain rule explanations
J A Rossiter
http://controleducation.group.shef.ac.uk/mathematics.html
Slides by Anthony Rossiter
2
Introduction
• The first 5 videos introduced differentiation from
first principles and product and quotient rules.
• The next very commonly used rule is the chain
rule (sometimes function of a function rule).
• This video explains the origins of the rule so
students can understand it better.
Students can skip these explanations and go
straight to resources 11 if they want to get straight
into using the rule.
Slides by Anthony Rossiter
3
What is a function of a function
Consider the following two simple functions.
y  f ( x)  sin( x); z  g ( x)  x
2
We can create two different functions from these
as follows:
h( x)  f ( g ( x))  sin( g ( x))  sin( x );
2
k ( x)  g ( f ( x))  [sin( x)]  sin x
2
2
The terminology ‘function of a function’
should be clear from this.
Slides by Anthony Rossiter
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Objective
Are there simple methods for obtaining the
derivative of a function of a function, or indeed,
function of a function of a function … .
We will begin from the basic definition of
differentiation given earlier.
dy
y( x  x)  y( x)
 lim x0
dx
( x  x)  x
Slides by Anthony Rossiter
Applying definition to function of a
function
Consider a simple function of a function.
y  f ( x)  sin( x);
z  g ( x)  x
2
h( x)  f ( g ( x))  sin( x );
Now apply the definition.
dh
h( x  x)  h( x)
 limx0
dx
( x  x)  x
dh
f ( g ( x  x))  f ( g ( x))
 limx0
dx
x
Slides by Anthony Rossiter
2
5
Applying definition to function of a
function
Consider the simple approximation of g(x+δx).
dg 

limx0  g ( x  x)  g ( x)  x 
dx 

dg 

g  x 
dx 

Substitute into the derivative of the previous page.
dh
f ( g ( x  x))  f ( g ( x))
 limx0
dx
x
dh
f ( g ( x)  g )  f ( g ( x))
 limx0
dx
x
Slides by Anthony Rossiter
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Applying definition to function of a
function
Next, consider the derivative of the function f(x)
only.

df  Use g as this
limg 0  f ( g  g )  f ( g )  g  is argument

dg

in function f.
Substitute into the derivative of the previous page.
dh
f ( g  g )  f ( g )
 limx0
dx
x
df
df
f ( g )  g  f ( g )
g
dh
dg
dg
 limx0
 limx0
dx
x
x
Slides by Anthony Rossiter
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Applying definition to function of a
function
Recall definition of δg from 2 pages ago:
dh
 limx0
dx
df
g
dg
;
x
AND HENCE WITH
dh
 limx0
dx
Slides by Anthony Rossiter
dg
g  x
dx
h( x)  f ( g ( x))
df dg
x
df dg
dg dx

x
dg dx
This is called the
chain rule.
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9
Differentiation with multiple functions
What if functions are nested multiple times over. It
is straightforward to show.
dh df dg dk
h( x)  f ( g (k ( x))) 

dx dg dk dx
dh df dg dk dw
h( x)  f ( g (k ( w( x)))) 

dx dg dk dw dx
This is called the chain rule as each function is differentiated in
order (like a chain), where the order is linked to the nesting.
Slides by Anthony Rossiter
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Example 1
Find the derivative of h(x).
y  f ( x)  sin( x); h( x)  f ( g ( x))  sin( x 2 );
2
z  g ( x)  x
From the chain rule:
dh df dg

dx dg dx
df
f  sin( g ) 
 cos( g ) g  x 2  dg  2 x
dg
dx
dh
2
 cos( g )2 x  2 x cos( x )
dx
Slides by Anthony Rossiter
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Example 2
Find the derivative of h(x).
f ( g )  tan(2 g ); g ( w)  log(3w); wx   e  1
2x
h( x)  f ( g ( w( x)))  tan(2log(3(e  1)));



w
From the chain rule:



2x
g
df
f  tan(2 g ) 
 2 sec 2 (2 g )
dg
dg 1
g  log(3w) 

dw w
dw
w  e 1 
 2e 2 x
dx
2x
Slides by Anthony Rossiter
dh df dg dw

dx dg dw dx
dh
1 2x
2
 2 sec (2 g ). .2e
dx
w
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Appendix
An easy short hand to remember for simple
functions of functions is given below [u=u(x)].
dh dh du
du
h  sin(u ) 

 cos(u )
dx du dx
dx
dh dh du
du
h  cosh(u ) 

 sinh(u )
dx du dx
dx
dh dh du 1 du
h  log(u ) 


dx du dx u dx
Slides by Anthony Rossiter
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Summary
• This brief resource has derived the chain rule for
differentiating functions that are nested
expressions of other functions.
• For example:
dh df dg dk dw
h( x)  f ( g (k ( w( x)))) 

dx dg dk dw dx
• The next video will give numerous examples.
Slides by Anthony Rossiter
Anthony Rossiter
Department of Automatic Control and
Systems Engineering
University of Sheffield
www.shef.ac.uk/acse
© 2016 University of Sheffield
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