Recall the definition of sin ✓ and cos ✓. 2⇡ radians = 360 .
Graph of sin ✓:
Graph of cos ✓:
Both are periodic functions with period 2⇡.
Section 3.6 – Trigonometric Derivatives
d
Find sin x graphically and algebraically.
dx
Solution: graphically
d
sin x = cos x
dx
Solution: algebraically
d sin(x + h) sin(x)
sin x = lim
dx h!0 h
sin x cos h + sin h cos x sin x
= lim
h!0
✓ h ◆ ✓ ◆
cos h 1 sin h
= sin x · lim + cos x · lim
h!0 h h!0 h
= sin x · 0 + cos x · 1 = cos x
Where we used
p
cos h 1 1 sin2 h 1
lim = lim
h!0 h h!0 h
p p
( 1 sin2 h 1)( 1 sin2 h + 1)
= lim p
h!0 h ( 1 sin2 h + 1)
sin h sin h 0
= lim · lim p =1· p = 0.
h!0 h h!0 1 sin2 h + 1 1+1
Theorem (Derivatives of Trigonometric functions)
d
sin x = cos x
dx
d
cos x = sin x
dx
d 1
tan x = sec2 x =
dx cos2 x
d 1
cot x = csc2 x =
dx sin2 x
d
sec x = sec x tan x
dx
d
csc x = csc x tan x
dx
Here: sec2 x = (sec x)2 , etc.
Note the pattern: The derivatives of the “co”-trigonometric
functions all have minus (-) signs.
For sin x, we showed already how to get the derivative.
For cos x this can be done similarly or one uses the fact that the
cosine is the shifted sine function.
The remaining trigonometric functions can be obtained from the
sine and cosine derivatives.
d
Example: Find tan x.
dx
Solution:
✓ ◆
d d sin x
tan x = (use quotient rule)
dx dx cos x
d d
( sin x) · cos x sin x · ( cos x)
= dx dx
cos2 x
cos x cos x sin x( sin x)
=
cos2 x
2 2
= cos x +2 sin x = 12 = sec2 x
cos x cos x
Section 3.7 – The Chain Rule
Let x, u, y be quantities such that
u = g (x) y = f (u)
change in x ! change in u ! change in y
dy
dx = change in y with respect to x
du
dx = change in u with respect to x
dy
du = change in y with respect to u
Theorem (Chain Rule)
dy dy du
= ·
dx du dx
Leibniz version of the chain rule
dy dy du
= ·
dx x=a du u=g (a) dx x=a
The chain rule in function notation
u = g (x) y = f (u)
y = f (g (x)) = composition of f and g
dy dy du
Chain rule = · in Leibniz notation becomes
dx du dx
Theorem (Chain Rule)
d
f (g (x)) = f 0 (g (x)) · g 0 (x)
dx
The derivative of a composition is the derivation of the “outer”
function evaluated at the “inner” function times the derviative of
the “inner” function.
d
Example: Find sin(x 2 ).
dx
Solution:
d d 2
sin(x 2 ) = cos(x 2 ) · x = 2x cos(x 2 ).
dx dx
dy
Example: Identify the inner and outer functions and find dx .
a) y = sec(5x 2 ).
Solution:
y = sec(5x 2 ) = f (g (x)), f (u) = sec u (outer), g (x) = 5x 2 (inner).
f 0 (u) = sec u · tan u, g 0 (x) = 10x.
dy
= f 0 (g (x)) · g 0 (x)
dx ⇥ ⇤
= sec(5x 2 ) tan(5x 2 ) · 10x = 10x sec(5x 2 ) tan(5x 2 )
3
b) y = e 2x
Solution:
3
y = e 2x = f (g (x)), f (u) = e u (outer), g (x) = 2x 3 (inner).
f 0 (u) = e u , g 0 (x) = 6x 2 .
dy
= f 0 (g (x)) · g 0 (x)
dx
3 3
= e 2x · 6x 2 = 6x 2 e 2x
Theorem (General Power Rule)
d n n 1
g (x) = n g (x) · g 0 (x)
dx
This is a special case of the chain rule. The outer function is
y = u n , the inner is u = g (x).
d p d 1/3
x 3 + 2x
3
Example: x 3 + 2x =
dx dx
1
1 d 3
= 13 x 3 + 2x 3 · (x + 2x)
dx
2/3
= 13 x 3 + 2x · (3x 2 + 2)
d d ⇥ ⇤5
Example: sin5 (x x 2) = sin(x x 2)
dx dx
⇥ ⇤4 d
= 5 sin(x x 2) · sin(x x 2 )
dx
⇥ ⇤4 d
= 5 sin(x x 2 ) cos(x x 2 ) · (x x 2 )
dx
⇥ ⇤4
= 5 sin(x x 2 ) cos(x x 2 ) · (1 2x)
r q
p
Example: Compute the derivative of x2 + x 2 + x.
Solution:
r q
d p
x + x2 + x
2
dx
!
1 1 1
= q p · 2x + p p · 2x + p
2 2
p 2 x2 + x 2 x
2 x + x + x
1
2x+ 2p
p p
x
+ 2x
2 x 2+ x
= q p p
2 x2 + x2 + x
Homework: Compute the second derivative.
Homework solution
p p p p p p
60x 7/2 + 42x 2 7 x 2 + x + 4 x 2 + xx 3/2 16 x 2 + xx 3
p p ⇣ 2 p 2 p ⌘3/2
64x 3/2 x 3/2 +1 2
x + x x + x + x
