Derivatives 2.2St. Pius X High SchoolMs. HernandezAP Calculus IF06 Q1 Derivatives Unit

Some Differentiation Rules!

Yeah, we have some rules that make finding the derivative so much EASIER! Constant Power Constant Multiple Sum and Difference Sine and Cosine

Constant [ ] 0d cdx

[ 7] [7] 0d dydx dx

[ ( ) 0] [0] 0d df xdx dx

[ ( ) 3] [ 3] 0d ds tdt dt

2y k k is a constant ' 0y

Power 1[ ]n nd x nxdx

3 2[ ( ) ] 3d f x x xdx

1/ 3 2 / 332 / 3

1( ) '( ) 1/ 3( )3

g x x x g x xx

2 32 3

1 2[ ] 2dy dy x xx dx dx x

1 1 1 0[ ( ) ] ( ) ' (1) (1) (1)1 1d f x x x x xdx

Special case n=1 f(x)=x f’(x)=1

Constant Multiple [ ( )] '( )d cf x cf xdx

[ 2 ] 2(1) 2d y xdx

22

7 7[ ] 7 '( ) 7( 1)d y f x xdx x x

33 2 24 4 4[ ( ) ] ( ) ' (3 ) 4

3 3 3d tf t t t tdx

2 / 3 5/ 3

2 53 3

1 1 1 1[ ] ( 2 / 3)2 22 3

d dy x xdx dxx x

Sum and Difference [ ( ) ( )] '( ) '( )d f x g x f x g x

dx

4 3 1( ) 2 75

f x x x x 3 2 1'( ) 4 6 05

f x x x

327( ) 4 2 3

5xf x x x

221'( ) 8 2 05xf x x

Sine and Cosine[sin ] cosd x x

dx [cos ] sind x x

dx

2cos ' 2sinsin cos'2 2cos 3sin ' 1 sin 3cos

y x y xx xy y

y x x x y x x

TS 2 Rates of PVA = Position, Velocity, & AccelerationRATE OF CHANGERate = distance/timeThe function s gives the position of an object as a function of timeAverage velocity = change in distance

change in time Average velocity = s / t s = s(t +t) – s(t)

Find average velocity of a falling object

If a billiard ball is dropped from a height of 100 feet, its height s at time t is given by the position function

s = -16t2 + 100 s(t) is the position function of the

billiard ball measure in feet t = time measured in seconds 100 = “ORIGIN”AL HEIGHT aka Initial Height

Find the average velocity s(t) = -16t2 + 100 find average velocity over the time

period [1,2]s(1) = 84 feet and s(2) = 36 feetSo average velocity is –48 ft/s

36 84 482 1

st

Why is it 36 – 84 ?Why is the velocity negative?

Velocity function 0

( ) ( )( ) limt

s t t s tv tt

LOOKS LIKE THE DERIVATIVE!!!!!

So the velocity function is the Derivative of the position function !!!!!

YEAH!!!

Average velocity vs instant velocity

Average velocity between t1 and t2 is the slope of the secant lineInstantaneous velocity at t1 is the slope of your tangent line

Position function of a FREE falling objectNeglecting air resistance….s0 = initial height of the objectv0 = initial velocity of the objectg~ -32 ft/s2 or –9.8 m/s2

(acceleration due to gravity on earth)

20 0

1( )2

s t gt v t s

ExampleAt time t=0, a diver jumps from a platform diving board that is 32 feet above water. The position of the diver is given by the following position function:

Where s is measured in feet and t is measured in seconds.When does the diver hit the water?What is the diver’s velocity at impact?

2( ) 16 16 32s t t t

Example cont’d32 is the initial height (height of board above water)From the middle term, 16t, 16 is the initial velocity of the diver

To find the time t when the diver hits the water, let s = 0 and solve for t. If s = 0 then the position is 0, right b/c the diver HITS the water…..

2( ) 16 16 32s t t t

Example cont’dSo we let s = 0 and solve for t to find the time it takes for the diver to hit the water

2( ) 16 16 32s t t t

2

2

0 16 16 32

0 16( 2)0 16( 2)( 1)

1, 2

t t

t tt t

t t

t can not be negative… no negative time… this is not back to the future, ok? So t = 2 is the only logical answerAt t = 2 seconds, the diver hits the water –that’s fast!

Example cont’dNext, lets solve for the diver’s velocity at impact.We use t=2, b/c we just found out that’s the time it takes for the diver to hit the water and we want velocity at impact (like you know when the diver hits the water, duh)Remember, velocity is the derivate of position

2( ) 16 16 32s t t t

'( ) 32 16'(2) 64 16 48 /s t ts ft s