Quotient rule applied to trig, finding second derivatives, third derivatives, and so on!
Quotient rule applied to trig, finding second derivatives, third
derivatives, and so on!
Say what? Why would those be the derivatives? Let’s look at tan x:sin
tancos
xy x
x
2
cos ( [sin ]) sin ( [cos ])'
(cos )
d dx x x xdx dxyx
2 2
2 2
cos cos sin ( sin ) cos sin
cos cos
x x x x x x
x x
22
1sec
cosx
x
Now I want you students to figure out why the rest work. On the given sheet, show your group’s work. Step 1: Define as quotient. Step 2: DeriveStep 3: Simplify
Group 1: find y’ when y = csc x
Group 2: find y’ when y = sec xGroup 3: find y’ when y = cot x
2tan secd
x xdx
sin cosd
x xdx
sec sec tand
x x xdx
cos sind
x xdx
2cot cscd
x xdx
csc csc cotd
x x xdx
So how to memorize? Compare and contrast and come up with some patterns.
Ex 8 p.124 Differentiating Trigonometric Functions
Function Derivativea.y = x – tan x
b. y = x sec x
21 secdy
xdx
(sec tan ) sec (1)dy
x x x xdx
sec ( tan 1)x x x
Ex 9 p. 124 Different Forms of a Derivative
Differentiate both forms of
First form:
2nd Form:
Are these equivalent? Check it out!
1 coscsc cot
sin
xy x x
x
1 cos
sin
xy
x
2 2
2 2
sin (sin ) (1 cos )(cos ) sin cos cos'
sin sin
x x x x x x xy
x x
2
1 cos
sin
x
x
csc coty x x
2' csc cot cscy x x x
Much of the work in calculus comes AFTER taking the derivative. Characteristics of a simplified form?Absence of negative exponents
Combining of like terms
Factored forms
Higher-Order Derivatives
Just as velocity is the derivative of a position function, acceleration is the derivative of a velocity function.
s(t) Position Function. . . .v(t) = s’(t) Velocity Function. . . .
a(t) = v’(t) = s”(t) Acceleration Function.
a(t) is the second derivative of s(t) – which is the derivative of a derivative!
Notations for higher-order derivatives:
2 2
2 2
3 3
3
ddyFirst derivative: y' '( ) ( )dx dx
d2nd derivative: y'' ''( ) ( )
dx
d3rd derivative: y''' '''( )
d
f x f x
d yf x f x
dx
d yf x
dx
3
4 4(4) (4)
4 4
n n(n) ( )
n
( )x
d4th derivative: y ( ) ( )
dx
dnth derivative: y ( ) ( )
dxn
n
f x
d yf x f x
dx
d yf x f x
dx
Ex 10 p. 125 Finding Acceleration Due to GravityBecause the moon has no atmosphere, a falling object on the moon hits no air resistance. In 1971, astronaut David Scott showed that a hammer and a feather fell at the same rate on the moon. 2( ) 0.81 2s t t is the position
functionwhere s(t) is the height in meters and t is time in seconds. What is the ratio of the Earth’s gravitational force to the moon’s?To find acceleration due to gravity on moon, differentiate twice.2( ) 0.81 2 position
'( ) 1.62 velocity
''( ) 1.62 acceleration
s t t
s t t
s t
2
2
Earth's gravitational force 9.8 m/sec
Moon's gravitational force 1.62 m/sec
6.05
Assignment2.3b p. 126 #45, 61, 73-78, 83-87 odd, 93-101 odd