# M119 Notes_ Lecture 2

Document Sample

```					3.3 Chain Rule
If y  f (u);u  g ( x) , then if we want to take the derivative of y with respect to x,
we need to first go through u.
dy dy du
        
d
 f ( g ( x))  f ' ( g ( x)) * g ' ( x)
dx du dx         dx

Ex. If we want the derivative of y  f ( x)  ( x  1)99 , what is a good substitution
(u=________)?

Find the derivative of:
1)   y  s2  1
2)   P  50e0.6t
3)   C  12 (3q 2  5)3
4)   f ( x)  ln(1  e  x )

Ex. If \$5000 is deposited in a bank that pays 3% annually, compounded continuously,
how fast is the money growing at 10 years?

Ex. Find the equation of the line tangent to y  [ln( x  3)]4 at x  2
3.4 Product and Quotient rule
Let’s review all the rules we already know:
d n
( x )  nx n 1
dx
d x
(e )  e x
dx
d x
(a )  (ln a )a x
dx
d           1
(ln x) 
x          x
dy dy du

dx du dx

While adding or subtracting terms or multiplying/dividing by a constant doesn’t effect the
derivative rules, multiplying or dividing two terms that both have variables changes the
rules:
d               dv     du
(uv)  u  v              uv'vu'
dx              dx     dx
du      dv                   or in terms of f and g
v      u
d u                            vu'uv'
   dx 2 dx 
dx  v            v                 v2
d
( f ( x) g ( x))  f ( x) g ' ( x)  g ( x) f ' ( x)
dx
d  f ( x)  g ( x) f ' ( x)  f ( x) g ' ( x)
         
dx  g ( x) 
                      g ( x)2
A hint on remembering the quotient rule: start with the bottom on both the top and
bottom, and then get the square on the bottom. Then just finish the top like the product
rule but with subtraction.

Differentiate:
A) y  (3 x  1) x 2

B) y  xe x
2x
C) y 
x 1

3
D) y 
x

If the quantity sold is given in terms of price is given by q  800 e .01 p
A) Write revenue as a function of price
B) Find the marginal revenue
C) Find the revenue and marginal revenue when the price is \$12 (including units)
s ( x )  f ( g ( x))
f (4)  8     f ' ( 4)  3
u ( x)  g ( f ( x))
g (4)  2     g ' ( 4 )  1
If                              and v ( x)  f ( x ) g ( x) then find
f (2)  11    f ' ( 2)  6
g ( x)
g (8)  2    g ' (8)  5       w( x) 
f ( x)

A)   s(4)
B)   s' (4)
C)   u ' ( 4)
D)   v' (4)
E)   w' (4)
Practice problems: Differentiate
A) y  (3 x  1) x 2
B) y  13(4t  3) 4
C) f ( z )  15 z  8e 5 z
D) y  t 6 (2t  10 ) 3
x2  4
E) z 
x2

Find the second derivative of y  ln( x 2  4)

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 4 posted: 9/15/2011 language: English pages: 5