The Chain Rule

1
The Chain Rule

• Working on the Chain Rule

2
Review of Derivative Rules

• Using Limits

3
Power Rule

• If f(x)

4
Product Rule
5
Quotient Rule
6
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

7
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.

8
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
  

9
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

10
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( )

11

• A. f(x) cos(x-1) f(x) -sin(x-1)
• B. f(x) cos(2x) f(x) -2sin(2x)
• C. f(x) sin(2 ) f(x) 4xcos(2 )

12
Combining Chain Rule

• Let f(x)sin(2x)cos(2x). Find f(x)

13
Combine product rule and chain rule
(2x)
(2x)

• 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)

14
Combine quotient rule and chain rule