by: susana cardona & demetri cheatham © cardona & cheatham 2011
TRANSCRIPT
DIFFERENTIATION A.K.A DERIVATIVE
Slope of a tangent line
Six different techniques: Chain rule, product rule, Quotient rule, E.T.A, Implicit differentiation and Logs.
Chain Rule
Bring exponent down in front of the variable, if it’s a coefficient multiply exponent. Then subtract one from the exponent and go back in and take a derivative.
1
( )
( )
n
n
f x ax
f x anx
Product Rule
First write the problem times derivative of the second problem plus write the second problem times the derivative of the first problem.
FDS+SDF
Example
2 3 3 7(5 1) (2 4)x x 2 3 3 6 2 3 7 2 4(5 1) (7)(2 4) (6 ) (2 4) ( 3)(5 1) (10 )x x x x x x
2 4( ) ( 3) (3 1)f x x x 2 3 4( ) ( 3) (4)(3 1) (3) (3 1) (2)( 3)(1)f x x x x x
( )f x
( )f x
Quotient Rule
Write the bottom times the derivative of the top minus write the top times the derivative of the bottom over the bottom squared
2
BDT TBD
B
Example
3
4
(5 1)
(2 1)
x
x
4 2 3 3
8
(2 1) (3)(5 1) (5) (5 1) (4)(2 1) (2)
(2 1)
x x x x
x
( )f x
( )f x
Solution
2 4 2 2 2 2 4 3
2 4 2
(6 2 )(2)(3 5 )(6 5) (3 5 ) (6 2 )(12 8 )
(6 2 )
x x x x x x x x x x x
x x
( )f x
ETA A.K.A Exponent, Trig, Angle
Bring down exponent, multiply coefficient if there’s one, and write the trig and the angle times the derivative of the trig times the derivative of the angle
Example
1.
2.
2(sin 3 )d
xdx
3(cos (sin ))d
xdx
23cos (sin ) ( sin(sin )) (cos )x x x
co2s s33 3in x
E
x
T A
Implicit
•Is almost the same as a chain rule but it includes x and y and the x’s and y’s can be separated
•
2 2 1
2 2 0
2
2
x y
dyx y
dxdy x
dx y
dy x
dx y
Solution
6 2 (1) 2 (1) 10 0
( 2 10 ) 2 6
2 6
2 10
3
5
dy dyx x y y
dx dxdy
x y y xdxdy y x
dx x y
dy y x
dx x y