The Chain Rule
• Working on the Chain Rule
Review of Derivative Rules
• Using Limits:
Power Rule
• If f(x) =
Product Rule
Quotient Rule
Why use the chain rule?
• The previous rules work well to take
derivatives of functions such as
• How do you best find a derivative of an
equation such as
The Chain Rule
• The chain rule is used to calculate derivatives
of composite functions, such as f(g(x)).
• Ex: Let f(x)=
and
• Therefore, f(g(x))=
• Obviously, it would be difficult to expand the
above function. The best way to calculate the
derivative is by use of the chain rule.
Chain Rule
(cont)
• The derivative of a composite function,
f(g)x)), is found by multiplying the derivative
of f(g(x)) by the derivative of g(x).
• Or, f’(g(x))(g’(x))
• In our example,
, we obtain
• This is the general power rule of the chain rule
Other applications of the chain rule
• To find f’(x) when f(x)=sin , f’(x)=
(cos )(2x)
• To find f’(x) when f(x)=
rewrite the
equation as
• Then, use the general power rule of the chain
rule to obtain
Trig and the Chain Rule
•
•
•
•
•
•
Let f(x)=sin u. f’(x)=(cos u)u’
Ex: f(x)=sin2x, f’(x)=cos2x(2)=2cos2x
Find the following derivatives:
A. f(x)=cos(x-1)
B. f(x)=cos(2x)
C. f(x)=sin( 2x )
2
• A. f(x) = cos(x-1) f’(x) = -sin(x-1)
• B. f(x) = cos(2x) f’(x) = -2sin(2x)
• C. f(x) = sin(2x2 ) f’(x) = 4xcos(2x )
2
Combining Chain Rule
• Let f(x)=sin(2x)cos(2x). Find f’(x)
(2x)
Combine product rule and chain rule
•
•
•
•
Let h(x)=sin(2x)cos(2x). Find h’(x)
From product rule, d/dx f(x)g(x)=
f’(x)g(x) + f(x)g’(x)
From above, if f(x)=sin(2x) and g(x)=cos(2x),
then f’(x)=2cos(2x) and g’(x)=-2sin(2x)
• Therefore, h’(x)=(2cos(2x))(cos(2x)) +
(sin(2x))(-2sin(2x)) =
(2x)
(2x)
Combine quotient rule and chain rule