Theorem: The less you know,
the more money you make

Proof:
We know that
a) Time is Money (T=M)
b) Knowledge is Power (K=P)
and from Physics
c) Power = Work / Time (P=W/T)

By substitution, K = W/M
Rearrange the equation and M = W/K, or
Money = Work/Knowledge

From this equation, it follows that as knowledge goes to
0, money goes to infinity.

Consider the function
y  sin  
 slope
We could make a graph of the slope:



2
1
0
1
Now we connect the dots!
0

2
The resulting curve is a cosine curve.

1
0
d
sin  x   cos x
dx

We can do the same thing for y  cos  
 slope


The resulting curve is a sine curve that has
been reflected about the x-axis.

2
0

2

0
1
0
1
0
d
cos  x    sin x
dx

Derivative of y=sinx

Use the definition of the derivative
lim f ( x  h)  f ( x)
y' 
h0
h
To prove the derivative of y=sinx is y’=cosx.
Hints:
sin( x  h)  sin x cosh  cos x sinh
lim cosh  1
0
h0 h
Derivative of y=sinx
lim f ( x  h)  f ( x)
y' 
h0
h
lim sin( x  h)  sin x

h0
h
lim sin x cosh  cos x sinh  sin x

h0
h
lim sin x(cosh  1)  cos x sinh

h0
h
lim 
cosh  1
sinh 

sin x 
 cos x 

h  0
h
h 

lim
h0
sin x 
lim cosh  1 lim
lim sinh

cos x 
h0 h
h0
h0 h
 sin x  0  cos x 1
Shortcut: y’=cosx
The proof of the d(cosx) = -sinx is almost identical
Derivative of y=tanx

Use the quotient rule to show the
derivative is y’=sec2x
sin x
y
cos x
HI  sin x dHI  cos x
HO  cos x dHO   sin x
cos 2 x  sin 2 x
y' 
cos 2 x
1

cos 2 x
Shortcut: y’=sec2x
The proof of the
d(cotx) = -csc2x
is almost identical
Derivative of y=secx

Use the quotient rule to show the
derivative is y’=secxtanx
1
y
cos x
HI  1
dHI  0
HO  cos x dHO   sin x
cos x 0  1 ( sin x)
y' 
cos 2 x
1 sin x

cos x cos x
Shortcut: y’=secxtanx
The proof of the
d(cscx) = -cscxcotx
is almost identical
Summary of trig derivatives
d
sin x  cos x
dx
d
cot x   csc 2 x
dx
d
cos x   sin x
dx
d
sec x  sec x  tan x
dx
d
tan x  sec 2 x
dx
d
csc x   csc x  cot x
dx

Now for a worksheet