Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
A Zoo of Trig Derivatives
Building the Derivative Machine, Part III
Peter A. Perry
University of Kentucky
September 30, 2016
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
Derivatives of Sine and Cosine
• Remember e x
• Three Key Ingredients
• Derivatives of Sine and Cosine
• A Zoo of Trig Function Derivatives
A Zoo of Trig Derivatives
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
Remember e x
Recall that e is the unique base with
eh − 1
= 1.
h →0
h
lim
We used this fact to find the derivative of f (x ) = e x .
f (x + h ) − f (x )
h
x
+
h
e
− ex
= lim
h →0
h
ex eh − ex
= lim
h →0
h
h
e −1
= e x lim
h →0
h
x
=e
f 0 (x ) = lim
h →0
A Zoo of Trig Derivatives
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
A Zoo of Trig Derivatives
The Next Step
Next, we’ll find derivatives of the trigonometric functions. Since all
of these functions are built out of sine and cosine, we first need to
find
d
d
(sin x ) ,
(cos x ) .
dx
dx
To do this we’ll need some basic limits. Remember that the key to
finding (d /dx )e x was to know the limit
eh − 1
h →0
h
lim
To find the derivatives of cosine and sine, we’ll need to know the
following limits:
sin h
= 1,
h →0 h
lim
cos(h ) − 1
=0
h →0
h
lim
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
A Zoo of Trig Derivatives
Two Little Limits, Part I
sin h ≤ h
h ≤ tan h
P (1, tan h)
P (cos h, sin h)
h
0
so
sin h
≤ 1,
h
h≤
sin h
cos h
or
1
sinh
≤1
h
Putting the “squeeze” on we get
cos h ≤
lim
h →0
sin h
=1
h
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
Two Little Limits, Part II
Using the fact that
sin h
=1
h →0 h
lim
(cos h − 1)(cos h + 1)
cos h − 1
= lim
h →0
h →0
h
h (cos h + 1)
2
cos h − 1
= lim
h→0 h (1 + cos h )
lim
− sin2 h
h→0 h (1 + cos h )
sin h − sin h
·
= lim
h →0 h
1 + cos h
=0
= lim
A Zoo of Trig Derivatives
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
A Zoo of Trig Derivatives
Sine and Cosine Addition Formulas
We’ll need the addition formulas
sin(α + β) = sin α cos β + cos β sin α
cos(α + β) = cos α cos β − sin α sin β
because we need to compute
lim
h →0
sin(x + h ) − sin x
,
h
lim
h →0
cos(x + h ) − cos(x )
h
These formulas will allow us to reduce the derivative calculation to
our two special limits
lim
h →0
sin h
,
h
lim
h →0
cos(h ) − 1
h
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
A Zoo of Trig Derivatives
The Derivative of the Sine Function
Addition formula for sine: sin(α + β) = sin α cos β + cos β sin α
sin(x + h ) − sin(x )
sin x cos h + cos x sin h − sin(x )
= lim
h →0
h →0
h
h
cos h − 1
sin h
= lim sin x
+ cos x
h →0
h
h
cos h − 1
sinh
= sin x lim
+ cos x lim
h →0
h →0 h
h
= sin x · 0 + cos x · 1
lim
= cos x
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
A Zoo of Trig Derivatives
The Derivative of the Cosine Function
Addition formula for cosine: cos(α + β) = cos α cos β − sin β sin α
cos(x + h ) − cos(x )
cos x cos h − sin x sin h − cos x
= lim
h →0
h →0
h
h
cos h − 1
sin h
= lim cos x
− sin x
h →0
h
h
cos h − 1
sin h
= cos x · lim
− sin x · lim
h →0
h →0 h
h
= cos x · 0 − sin x · 1
lim
= − sin x
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
Graphs and Derivatives
Which is f (x ) = cos(x ) and which is f 0 (x ) = − sin(x )?
Where is f 0 (x ) = 0?
Where is f increasing, and where is it decreasing?
−π
−π/2
0
π/2
π
A Zoo of Trig Derivatives
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
Graphs and Derivatives
f (x ) = cos(x ) and f 0 (x ) = − sin x
Where is f 0 (x ) = 0?
Where is f increasing, and where is it decreasing?
−π
−π/2
0
π/2
π
A Zoo of Trig Derivatives
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
Graphs and Derivatives
f (x ) = cos(x ) and f 0 (x ) = − sin x
Where is f 0 (x ) = 0? x = −π, x = 0, x = π
Where is f increasing, and where is it decreasing?
m=0
m=0
−π
m=0
−π/2
0
π/2
π
A Zoo of Trig Derivatives
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
A Zoo of Trig Derivatives
Graphs and Derivatives
f (x ) = cos(x ) and f 0 (x ) = − sin x
Where is f 0 (x ) = 0? x = −π, x = 0, x = π
Where is f increasing, and where is it decreasing? Increasing
(−π, 0), decreasing (0, π )
m=0
m=0
−π
m=0
−π/2
0
π/2
π
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
Graphs and Derivatives
Which is f (x ) = sin(x ) and which is f 0 (x ) = cos(x )?
Where is f 0 (x ) = 0?
Where is f increasing, and where is it decreasing?
−π
−π/2
0
π/2
π
A Zoo of Trig Derivatives
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
Graphs and Derivatives
f (x ) = sin(x ) and f 0 (x ) = cos x
Where is f 0 (x ) = 0?
Where is f increasing, and where is it decreasing?
−π
−π/2
0
π/2
π
A Zoo of Trig Derivatives
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
Graphs and Derivatives
f (x ) = sin(x ) and f 0 (x ) = cos x
Where is f 0 (x ) = 0? x = −π/2, x = π/2
Where is f increasing, and where is it decreasing?
m=0
m=0
−π
−π/2
0
π/2
π
A Zoo of Trig Derivatives
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
A Zoo of Trig Derivatives
Graphs and Derivatives
f (x ) = sin(x ) and f 0 (x ) = cos x
Where is f 0 (x ) = 0? x = −π/2, x = π/2
Where is f increasing, and where is it decreasing? Increasing
(−π/2.π/2), decreasing (−π, −π/2) ∪ (π/2, π )
m=0
m=0
−π
−π/2
0
π/2
π
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
A Zoo of Trig Derivatives
Mathematics, the Science of Patterns
If f (x ) = sin x, find f 0 (x ), f 00 (x ), f 000 (x ), and f (4) (x ).
Find f (8) (x ) if f (x ) = sin x.
A. sin x
B. − sin x
C. cos x
D. − cos x
E. tan x
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
A Zoo of Trig Derivatives
Mathematics, the Science of Patterns
If f (x ) = sin x, find f 0 (x ), f 00 (x ), f 000 (x ), and f (4) (x ).
f 0 (x ) = cos x
f 000 (x ) = − cos x
Find f (8) (x ) if f (x ) = sin x.
A. sin x
B. − sin x
C. cos x
D. − cos x
E. tan x
f 00 (x ) = − sin x
f (4) (x ) = sin x
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
A Zoo of Trig Derivatives
What About the Other Trig Functions
We can get the rest of the trig functions by using
sin2 x + cos2 x = 1
and
d
(sin x ) = cos x,
dx
d
(cos x ) = − sin x
dx
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
A Zoo of Trig Derivatives
What About the Other Trig Functions
We can get the rest of the trig functions by using
sin2 x + cos2 x = 1
and
d
(sin x ) = cos x,
dx
d
d
(tan x ) =
dx
dx
d
(cos x ) = − sin x
dx
sin x
cos x
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
A Zoo of Trig Derivatives
What About the Other Trig Functions
We can get the rest of the trig functions by using
sin2 x + cos2 x = 1
and
d
(sin x ) = cos x,
dx
d
d
(tan x ) =
dx
dx
=
d
(cos x ) = − sin x
dx
sin x
cos x
cos x cos x − sin x · (− sin x )
cos2 x
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
A Zoo of Trig Derivatives
What About the Other Trig Functions
We can get the rest of the trig functions by using
sin2 x + cos2 x = 1
and
d
(sin x ) = cos x,
dx
d
d
(tan x ) =
dx
dx
d
(cos x ) = − sin x
dx
sin x
cos x
=
cos x cos x − sin x · (− sin x )
cos2 x
=
cos2 x + sin2 x
cos2 x
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
A Zoo of Trig Derivatives
What About the Other Trig Functions
We can get the rest of the trig functions by using
sin2 x + cos2 x = 1
and
d
(sin x ) = cos x,
dx
d
d
(tan x ) =
dx
dx
d
(cos x ) = − sin x
dx
sin x
cos x
=
cos x cos x − sin x · (− sin x )
cos2 x
=
cos2 x + sin2 x
cos2 x
= sec2 x
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
A Zoo of Trig Derivatives
Graphs and Derivatives
Which graph is f , which graph is f 0 , and what’s f ?
−π −π/2
0
π/2
π
−π −π/2
0
π/2
π
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
A Zoo of Trig Derivatives
Graphs and Derivatives
Which graph is f , which graph is f 0 , and what’s f ?
−π −π/2
0
π/2
f (x ) = tan x
π
−π −π/2
0
π/2
f 0 (x ) = sec2 x
π
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
A Zoo of Trig Derivatives
All The Trig Derivatives You’ll Ever Need
Derivatives of Trig Functions
d
(sin x ) = cos x
dx
d
(cos x ) = − sin x
dx
d
(tan x ) = sec2 x
dx
d
(cot x ) = − csc2 x
dx
d
(sec x ) = sec x tan x
dx
d
(csc x ) = − csc x cot x
dx
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
A Zoo of Trig Derivatives
Let f (t ) = t + sin(t ). Find all values of t for which f 0 (t ) = 0
A. All even multiples of π
B. All odd multiples of π
C. All even multiples of π/2
D. All odd multiples of π/2
E. None of the above
Remember e x
Three Key Ingredients
Derivatives of Sine and Cosine
A Zoo of Trig Derivatives
Let f (t ) = t + sin(t ). Find all values of t for which f 0 (t ) = 0
f 0 (t ) = 1 + cos(t ) so f 0 (t ) = 0 if t = . . . − 3π, −π, π, 3π, . . .
A. All even multiples of π
B. All odd multiples of π
C. All even multiples of π/2
D. All odd multiples of π/2
E. None of the above