The Chain Rule
by CHED on June 17, 2017
lesson duration of 3 minutes
under Basic Calculus
generated on June 17, 2017 at 04:44 pm
Tags: Higher-Order Derivatives and the Chain Rule, Derivatives
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Generated: Jun 18,2017 12:44 AM
The Chain Rule
( 3 mins )
Written By: CHED on June 26, 2016
Subjects: Basic Calculus
Tags: Higher-Order Derivatives and the Chain Rule, Derivatives
Resources
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Content Standard
Basic concepts of derivatives
Performance Standard
Formulate and solve accurately situational problems involving extreme value
Formulate and solve accurately situational problems involving related rates
Learning Competencies
Solve problems using the Chain Rule
Illustrate the Chain Rule of differentiation
Introduction
1 mins
Consider the following functions:
(a)
(b) y = sin2x
sin2x
Teaching Tip
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Ask the students to find the derivatives f'(x
'(x) and y' of the functions above, before continuing with your lecture.
Expect some students to use the Power Rule (even when it is not applicable):
(This is incorrect!)
Some may expand the expression first to get
before differentiating:
( This is correct!)
Some may even use product rule and first write the function
Hence,
Ask them if the last two (correct) methods will be doable if
.
On the
the other
other hand,
hand, for
for the
the function
function y = sin2
sin2x
x, ,some
someofofthem
themmay
mayhave
haveused
usedaatrigonometric
trigonometricidentity
identitytotofirst
firstrewrite
rewrite y
On
into
y = sin2x
sin2x = 2sinx cos x.
In this case
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thislesson
lessonstudents
studentswill
willlearn
learna arule
rulethat
thatwill
willallow
allowthem
themtotodifferentiate
differentiatea agiven
givenfunction
functionwithout
withouthaving
havingtotoperform
perform
InInthis
any preliminary multiplication, or apply any special formula.
Lesson Proper 1 mins
The Chain Rule below provides for a formula for the derivative of a composition of functions.
Letf be
f bea afunction
functiondifferentiable
differentiableatatc cand
andletletg gbe
bea afunction
functiondifferentiable
differentiableatatf(c).
f(c).Then
Thenthe
the
Theorem 9 (Chain Rule). Let
composition
is differentiable at c and
Remark 1:
Another way
way to
to state
state the
the Chain
Chain Rule
Rule is
is the
the following:
following: IfIf y is a differentiable function of u defined by y = f(u)
1 : Another
and u is a differentiable function of x defined by u = g(x), then y is a differentiable function of x, and the derivative of y
with respect to x is given by
In words,
derivative
a composition
of functions
is the
derivative
of the
outer
function
evaluated
at the
inner
In words,
thethe
derivative
of aofcomposition
of functions
is the
derivative
of the
outer
function
evaluated
at the
inner
function, times the derivative of the inner function.
EXAMPLE 1:
(a)Recall
Recallour
ourfirst
firstillustration
illustration
(a)
using the Chain Rule.
Solution.
We can
can rewrite
rewrite y =
Solution . We
Find f'(x
.. Find
'(x)
as y = f(u) =
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where where
function of x. Using the Chain Rule, we have
,
a,
(b) For the second illustration, we have y = sin(2x)
sin(2x).. Find y' using the Chain Rule.
Solution. We can rewrite y = sin(2x
sin(2x) as y = f(u) wheref
wheref(u) = sinu
sinu and u = 2x
2x.
Hence,
(c) Find
if
Solution. Notice that we can write
, where a is a real number.
as
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a
differe
differentiabl
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K-12 Teacher's Resource Community
. Applying the Chain Rule, we have
Now, we
suppose
wefind
want
find the derivative
a poweroffunction
That
we are interested
in
Now, suppose
want to
thetoderivative
of a powerof function
x. That ofis,x.we
areis,interested
in
To derive
a formula
for this,
we let
. To. derive
a formula
for this,
we let
differentiable function of x given by u = f(x). Then by the Chain Rule,
where u isis aa
where
Thus,
GENERALIZED POWER RULE.
. This is called the
EXAMPLE 2:
(a) What is the derivative of y =
?
Solution.
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(b) Find
where y =
Solution.
(c) Find
where y =
.
Solution.
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(d) Differentiate
Solution.
(e) Consider the functions
and
. Find
Solution. By the Chain Rule, we have
.
where
= 6u
6u + 4 and
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= 2x
2x.
Thus,
Exercises 1 mins
1. Use the Chain Rule to find
in terms of x.
x.
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2. Solve for
and simplify the result.
3. Find the derivatives of the following functions as specified:
4. Find the first and second derivatives of the following:
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5. Find
If y = sin
6. Find
, given h(x) = sinx
sinx cos 3x
3x.
7. If
, then find
for all
For problems (8) and (9), please refer to the table below:
8. Use the table of values to determine (f
(f o g)(x
)(x) and (f
(f o g)'(x
)'(x) at x = 1 and x = 2.
9. Use the table of values to determine (f
(f o g)(x
)(x) and (f
(f o g)'(x
)'(x) at x = -2,-1 and 0.
10. Show that z" + 4z
4z' + 8z
8z = 0 if z =
(sin 2x
2x + cos 2x
2x).
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11. If f(x) =
and g(x) =
and g'(
g'(x) = 3 The Chain Rule multiplies derivatives to get
and its derivative is NOT
, then f'(x
'(x) = 4
. But f (g
(g(x)) =
=
. Where is the flaw?
Download Teaching Guide Book 0 mins
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