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Sample

The Chain Rule

Finding derivatives of compositions of functions

Basic Compositions of Functions

You have already worked with derivative rules for many functions.
◼
Find the derivatives for
sin(x)
,
5
x
,
ln(x)
, and
x
e
:
In[69]:=
D[Sin[x],x]
Out[69]=
cos(x)
In[19]:=
D[
5
x
,x]
Out[19]=
5
4
x
In[67]:=
D[Log[x],x](*notethatln(x)isinputasLog[x]inWL*)
Out[67]=
1
x
In[21]:=
D[
x

,x]
Out[21]=
x

A composition of functions is one function put inside another, such as
sin(2x)
.
The
2x
function is put inside the
sin(x)
function. Our first guess at the derivative of
sin(2x)
would probably be
cos(2x)
.
◼
What does the WL say is the derivative of
sin(2x)
?
In[24]:=
D[Sin[2x],x]
Out[24]=
2cos(2x)
Our guess was partly correct; the answer does contain
cos(2x)
, but also has an extra factor of
2
. Let’s try another example.
◼
Find is the derivative of
sin(
3
x
)
:
In[25]:=
D[Sin[
3
x
],x]
Out[25]=
3
2
x
cos(
3
x
)
Again, we expected to see
cos(
3
x
)
, but there is another “extra” factor of
3
2
x
.
Can we predict where this extra factor is coming from?
◼
Here are three more examples to help us with our prediction:
In[26]:=
D[Sin[
14
x
],x]
Out[26]=
14
13
x
cos(
14
x
)
In[27]:=
D[Sin[Log[x]],x]
Out[27]=
cos(log(x))
x
Note about logarithms in WL:
In the WL,
log(x)
is the same as
ln(x)
, the natural logarithm.
The WL uses
Log10[x]
for
log
10
(x)
In[28]:=
D[
7x

,x]
Out[28]=
7
7x

Notice that the argument of the “outside” function remains unchanged (the
7x
part in the last example)
◼
Predict each derivative, then calculate the answer:
DSin
x
,x
D[Sin[Tan[x]],x]
D[
5x

,x]
D
Sin[x]

,x
D[
4
(Sin[x])
,x]
You might describe this rule to a friend as:
​Take the derivative of the outside function, then multiply by the derivative of the inside function.
◼
Find the derivative of
g(
2
x
)
:
In[19]:=
D[g[
2
x
],x]
Out[19]=
2x
′
g
(
2
x
)
Using function notation, a composition of functions can be written as
f(g(x))
.
◼
Predict the derivative of
f(g(x))
and then calculate the answer:
In[30]:=
D[f[g[x]],x]
Out[30]=
′
g
(x)
′
f
(g(x))

Intermediate Examples

To apply the Chain Rule correctly, it is very important to identify the outside and inside functions.
Sometimes, it is helpful to rewrite a function (with algebra) to correctly see the outside and inside function. For example, consider rewriting the following function with parenthesis and a rational exponent.
◼
You can also use the Traditional Form of the square root:

Multiple Applications of the Chain Rule

Sometimes, compositions of functions are made up of more levels than f(g(x)).
◼
Find the derivative of f(g(h(x))):
Your answer might be in a slightly different format than an “expected” answer.
For example:
This is the way this answer might show up on the AP Exam or a college placement test.

Challenge Yourself

Try these problems and use WL to check your answers
◼
Use the slider below to see the pattern in the derivatives of the nested functions:
Further Explorations
Using the Chain Rule with the Product Rule
Using the Chain Rule with the Quotient Rule
Authorship information
Dan Uhlman
23 June 2017
uhlmand@tas.tw