Post on 21-Jan-2016
description
The Product Rule
Theorem. Let f and g be differentiable functions. Then the derivative of the product fg is
(fg) '(x) = f(x) g '(x) + g(x) f '(x)
In other words, first times the derivative of the second plus second times the derivative of the first.
Using the Product Rule
Example:2( ) (3 2 )(5 4 )h x x x x
2 2
2
2 2
2
(3 2 ) (5 4 ) (5 4 ) (3 2 )
(3 2 )(4) (5 4 )(3 4 )
(12 8 ) (15 20 12 16 )
24 4 15
d dx x x x x x
dx dx
x x x x
x x x x x
x x
Product Rule
Example:
3
3 3
3 2
2 3
( ) cos
cos cos
( sin ) cos (3 )
3 cos sin
f x x x
d dx x x xdx dx
x x x x
x x x x
Quotient Rule
Theorem. Let f and g be differentiable functions. Then the derivative of the quotient f/g is
2
( ) ( ) '( ) ( ) '( )
( ) ( )
d f x g x f x f x g x
dx g x g x
In other words, low d high minus high d low over low low.
Quotient Rule
ExampleFind the derivative of 2
5 2
1
xy
x
2
2 2
2 2
2 2
2 2
2 2
2
2 2
( 1)(5) (5 2)(2 )
( 1)
(5 5) (10 4 )
( 1)
5 5 10 4
( 1)
5 4 5
( 1)
x x x
x
x x x
x
x x x
x
x x
x
Rewriting Before Differentiating
Example:
2
2
2 2
2 2
2 2
2 2
2 2
2
2 2
13
( )5
13
( ) get rid of the fraction in the denom( 5)
3 1( )
5
( 5)(3) (3 1)(2 )'( )
( 5)
3 15 (6 2 )'( )
( 5)
3 15 6 2'( )
( 5)
3 2 15'( )
( 5)
xf xx
xx
f xx x
xf x
x
x x xf x
x
x x xf x
x
x x xf x
x
x xf x
x
Derivatives of Trigonometric Functions
2 2
sin cos cos sin
tan sec cot csc
sec sec tan csc csc cot
d dx x x x
dx dx
d dx x x x
dx dx
d dx x x x x x
dx dx
Proof of Derivative of Tangent
Considering sin
tancos
xx
x
2
2 2
2
2
2
(cos )(cos ) ((sin )( sin )tan
cos
cos sin
cos1
cos
sec
d x x x xx
dx x
x x
x
x
x
Differentiating Trig Functions
y = x – tan x
y = x sec x y’ = x(sec x tan x) + sec x (1)
= sec x (x tan x + 1)
2 2' 1 sec tany x x
Differentiating Trigonometric Functions
2
2 2
2
2
2 2
2
1 cos
sinsin (sin ) (1 cos )(cos )
'sin
sin cos cos
sin1 cos
sin1 cos
sin sin
csc csc cot
xy
xx x x x
yx
x x x
xx
xx
x x
x x x