Derivatives of the Trig and Exponential Functions
The derivative of sine
d
sin(x + h) sin(x)
sin x = lim
h!0
dx
h
sin(x) cos(h) + cos(x) sin(h)
= lim
h!0
h
sin(x)
The derivative of sine
d
sin(x + h) sin(x)
sin x = lim
h!0
dx
h
sin(x) cos(h) + cos(x) sin(h) sin(x)
= lim
h!0
h
sin(x)(cos(h) 1) + cos(x) sin(h)
= lim
h!0
h
The derivative of sine
d
sin(x + h) sin(x)
sin x = lim
h!0
dx
h
sin(x) cos(h) + cos(x) sin(h) sin(x)
= lim
h!0
h
sin(x)(cos(h) 1) + cos(x) sin(h)
= lim
h!0
h
cos(h) 1
sin(h)
= sin(x) lim
+ cos(x) lim
h!0
h!0
h
h
The derivative of sine
d
sin(x + h) sin(x)
sin x = lim
h!0
dx
h
sin(x) cos(h) + cos(x) sin(h) sin(x)
= lim
h!0
h
sin(x)(cos(h) 1) + cos(x) sin(h)
= lim
h!0
h
cos(h) 1
sin(h)
= sin(x) lim
+ cos(x) lim
h!0
h!0
h
h
Recall: cos(0) = 1 and sin(0) = 0
Near x = 0, sin(x) ⇡ x:
Graph of
sin(x)
x :
0.06
1
0.03
-0.09
-0.06
-0.03
0.03
0.06
-0.03
0.09
-8
-0.06
lim
x!0
sin(x)
=1
x
-6
-4
-2
2
4
6
8
Near x = 0, cos(x) ⇡ 1:
Graph of
cos(x) 1
:
x
1.025
0.5
1
-8
-6
-4
-2
2
-0.5
0.975
-0.025
0.025
lim
x!0
cos(x)
x
1
=0
4
6
8
The derivative of sine
d
sin(x + h) sin(x)
sin x = lim
h!0
dx
h
sin(x) cos(h) + cos(x) sin(h) sin(x)
= lim
h!0
h
sin(x)(cos(h) 1) + cos(x) sin(h)
= lim
h!0
h
cos(h) 1
sin(h)
= sin(x) lim
+ cos(x) lim
h!0
h!0
h
h
The derivative of sine
d
sin(x + h) sin(x)
sin x = lim
h!0
dx
h
sin(x) cos(h) + cos(x) sin(h) sin(x)
= lim
h!0
h
sin(x)(cos(h) 1) + cos(x) sin(h)
= lim
h!0
h
cos(h) 1
sin(h)
= sin(x) lim
+ cos(x) lim
h!0
h!0
h
h
= sin(x) ⇤ 0 + cos(x) ⇤ 1
= cos(x)
The derivative of cosine
d
cos(x + h) cos(x)
cos x = lim
h!0
dx
h
cos(x) cos(h) sin(x) sin(h)
= lim
h!0
h
cos(x)
The derivative of cosine
d
cos(x + h) cos(x)
cos x = lim
h!0
dx
h
cos(x) cos(h) sin(x) sin(h) cos(x)
= lim
h!0
h
cos(x)(cos(h) 1) sin(x) sin(h)
= lim
h!0
h
The derivative of cosine
d
cos(x + h) cos(x)
cos x = lim
h!0
dx
h
cos(x) cos(h) sin(x) sin(h) cos(x)
= lim
h!0
h
cos(x)(cos(h) 1) sin(x) sin(h)
= lim
h!0
h
cos(h) 1
sin(h)
= cos(x) lim
sin(x) lim
h!0
h!0
h
h
The derivative of cosine
d
cos(x + h) cos(x)
cos x = lim
h!0
dx
h
cos(x) cos(h) sin(x) sin(h) cos(x)
= lim
h!0
h
cos(x)(cos(h) 1) sin(x) sin(h)
= lim
h!0
h
cos(h) 1
sin(h)
= cos(x) lim
sin(x) lim
h!0
h!0
h
h
= cos(x) ⇤ 0
sin(x) ⇤ 1
The derivative of cosine
d
cos(x + h) cos(x)
cos x = lim
h!0
dx
h
cos(x) cos(h) sin(x) sin(h) cos(x)
= lim
h!0
h
cos(x)(cos(h) 1) sin(x) sin(h)
= lim
h!0
h
cos(h) 1
sin(h)
= cos(x) lim
sin(x) lim
h!0
h!0
h
h
= cos(x) ⇤ 0
=
sin(x)
sin(x) ⇤ 1
Does it make sense?
y = sin(x) :
y = cos(x) :
y=
sin(x) :
Examples
Calculate...
1.
d
dx
sin(2x) = 2 ⇤ cos(2x)
2.
d
dx
cos(3x +
=
d
dx
d
dx
sin(x) cos(x) = sin(x)( sin(x)) + cos(x) cos(x)
3.
p
x)
cos(3x + x 1/2 ) = (3 + 12 x
1
2
)
sin(3x + x 1/2 )
= cos2 (x)
Notice: sin(x) cos(x) = 12 sin(2x), and cos2 (x)
sin2 (x)
sin2 (x) = cos(2x).
Does your answer still make sense from this perspective?
4.
d
dx
sin(cos(x 2 + 2)) = cos(cos(x 2 + 2)) ⇤
d
dx
cos(x 2 + 2)
d
= cos(cos(x 2 + 2)) ⇤ ( sin(x 2 + 2)) ⇤
x2 + 2
dx
= cos(cos(x 2 + 2)) ⇤ ( sin(x 2 + 2)) ⇤ (2x)
On your own, fill in the rest of the trig functions:
1.
d
dx
tan(x) =
d sin(x)
dx cos(x)
= sec2 (x)
2.
d
dx
cot(x) =
d cos(x)
dx sin(x)
=
3.
d
dx
sec(x) =
d
1
dx (cos(x))
= sec(x) tan(x)
4.
d
dx
csc(x) =
d
1
dx (sin(x))
=
csc2 (x)
csc(x) cot(x)
The Derivative of y = e x
Recall!
e x is the unique exponential function whose slope at x = 0 is 1:
m=1
e 0+h e 0
eh 1
= lim
=1
h!0
h!0
h
h
lim
The Derivative of y = e x ...
lim
h!0
eh
1
h
=1
d x
e x+h e x
e = lim
h!0
dx
h
ex eh 1
= lim
h!0
h
h
e
1
= e x lim
h!0
h
= ex ⇤ 1
So
d x
e = ex
dx
Examples
Calculate...
1.
d 17x
dx e
2.
d x ln(3)
dx e
3.
d sin x
dx e
4.
p
d
x 2 +x
e
dx
= 17e 17x
= ln(3)e x ln(3)
= cos(x)e sin x
=
p
2x+1
x 2 +x
p
e
2 x 2 +x
Notice, every time:
d f (x)
e
= f 0 (x)e f (x)
dx
Other bases
We know
Notice:
d a
dx x
= ax a
1,
but what is
d x
dx a ?
x
ax = e ln(a ) = e xln(a)
So
Example:
d x
d x ln(a)
a =
e
= ln(a)e x ln(a) = ln(a) ⇤ ax
dx
dx
d x
dx 2
Sanity check:
= ln(2) ⇤ 2x
d x ?
dx e =
ln(e) ⇤ e x = 1 ⇤ e x
,
Derivative so far
Definition: f 0 (x) lim
h!0
f (x + h)
h
f (x)
Combining functions:
d
c ⇤ f (x) = c ⇤ f 0 (x)
dx
d
(f (x) + g (x)) = f 0 (x) + g 0 (x)
dx
d
(f (x) ⇤ g (x)) = f (x)g 0 (x)+g (x)f 0 (x)
dx
d
(f (g (x)) = f 0 (g (x))⇤g 0 (x)
dx
Basic functions:
f (x)
f 0 (x)
xa
ax a
1
sin(x)
cos(x)
cos(x)
sin(x)
ex
ex
Other functions:
f (x)
f 0 (x)
tan(x)
sec2 (x)
cot(x)
csc2 (x)
sec(x)
sec(x) tan(x)
csc(x)
csc(x) cot(x)
ax
ln(a)ax
Example
Compute the derivatives of
2
p
1. e 17x + x = e 17x
⇣
⌘
2 p
2. tan e 17x + x
2+
p
x
⇣
⇤ 17 ⇤ 2x +
1
p
2 x
⌘
⇣
⌘ ⇣
⇣
2 p
2 p
= sec2 e 17x + x ⇤ e 17x + x ⇤ 17 ⇤ 2x +
h
⇣
⌘i4
2 p
3. csc(x) ⇤ 3x + tan3 e 17x + x
1
p
2 x
⌘⌘