2.3 product and quotient rules

Question 1

If y=f(x)•g(x), what is dy/dx?

Answer 1

dy/dx=f’(x)•g(x)+g’(x)+f(x)
or vdu+udv, where y=u•v

Question 2

If y=f(x)/g(x), what is dy/dx?

Answer 2

dy/dx=[f’(x)•g(x)–g’(x)+f(x)]/[g(x)^2]
or (vdu-udv)/v^2, where y=u/v

Question 3

What are the derivatives of sin x and cos x?

Answer 3

d/dx[sin x]=cos x
d/dx[cos x]=–sin x

Question 4

What are the derivatives of tan x and cot x?

Answer 4

d/dx[tan x]=sec^2(x)
d/dx[cot x]=–csc^2(x)

Question 5

What are the derivatives of sec x and csc x?

Answer 5

d/dx[sec x]=secxtanx
d/dx[csc x]=–cscxcotx

Question 6

How should I derive the trig. derivatives?

Answer 6

sin x and cos x: use derivative definition and law of sin/cos addition, then separate into 2 separate limits
tan x and cot x: for tan use sin x/cos x in quotient rule, cot is the opposite (+ to -, sec to csc)
sec x and csc x: for sec use 1/cos x in quotient rule, csc is the opposite (+ to -, sec to csc, tan to cot)

Question 7

What are the laws of addition of sines and cosines?

Answer 7

sin(α +/- β)=sinαcosβ +/- cosαsinβ
cos(α +/- β)=cosαcosβ -/+ sinαsinβ

Question 8

What are the three special limits?

Answer 8

1. the limit of x approaching 0 of sinx/x=1 (lim x/sinx=1 as well b/c 1/1=1)
2. the limit of x approaching 0 of 1–cosx/x=0 (lim cosx–1/x=0 as well bc negative 0=0)
3. the limit of x approaching 0 of (1+x)^(1/x)=e