MATH 1300
Lecture Notes
October, 8 2013
1. Section 3.5 of HH - Trigonometric Functions
In this section we will find the derivatives of the following trigonometric functions:
d
(sin x) = cos x
dx
d
•
(cos x) = − sin x
dx
d
1
•
(tan x) =
= sec2 x
dx
cos2 x
d
•
(sec x) = sec x tan x
dx
•
(a) First, let’s try drawing a graph of the derivative of sin x. Below is a graph of
y = sin x, along with the tangent lines at x = 0, π, and 2π.
y
1
Π
x
3Π
Π
2Π
2
2
-1
Figure 1: y = sin x
We see that the derivative is zero at x = π2 and 3π
. Moreover, it looks like the
2
derivative has a maximum value of 1 when x = 0 and 2π, and a minimum value
of −1 when x = π.
1
Π
Π
3Π
2
2
2Π
-1
Figure 2: derivative of y = sin x
This looks a lot like the graph of y = cos x. It turns out, it is! From now on we
d
will assume that dx
(sin x) = cos x. You can find an informal geometric proof of
this fact on pages 142-143 of the text.
Warning! The above result is only valid when x is radians.
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(b) We saw in section 1.5 that cos x = sin(x + π2 ). Therefore, by the chain rule
d
d
π
(cos x) =
(sin(x + ))
dx
dx
2
π
= cos(x + )
2
= sin(x + π)
= − sin x.
(c) Now we can find the derivative of tan x.
d
d sin x
(tan x) =
dx
dx cos x
d
d
(cos x) · dx
(sin x) − (sin x) · dx
(cos x)
=
, by the quotient rule
cos2 x
(cos x)(cos x) − (sin x)(− sin x)
=
cos2 x
2
2
cos x + sin x
=
cos2 x
1
=
cos2 x
= sec2 x.
(d) Last, let’s find the derivative of sec x.
d
d
(sec x) =
((cos x)−1 )
dx
dx
d
= −(cos x)−2 · (cos x), by the chain rule
dx
−2
= −(cos x) (− sin x)
sin x
=
cos2 x
1
sin x
=
·
cos x cos x
= sec x tan x.
The derivatives of the other trigonometric functions, csc x and cot x, can be found
similarly.
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2. Examples
(a) Find the derivative of f (x) = sec(3x).
Solution. By the chain rule,
f 0 (x) = sec(3x) tan(3x) ·
d
(3x) = 3 sec(3x) tan(3x).
dx
(b) Find the derivative of f (x) = sin x cos x.
Solution. By the product rule,
d
d
(cos x) + (cos x) · (sin x)
dx
dx
= sin x(− sin x) + cos x cos x
= cos2 x − sin2 x.
f 0 (x) = (sin x) ·
(c) Find the derivative of f (x) = sin(cos x).
Solution. By the chain rule,
f 0 (x) = cos(cos x) ·
d
(cos x) = cos(cos x)(− sin x) = − sin x cos(cos x).
dx
(d) Find the derivative of f (x) = xetan x
Solution. By the product and chain rules,
d tan x
d
(e
) + (etan x ) · (x)
dx
dx
d
= x(etan x · (tan x)) + (etan x )(1)
dx
tan x
= x(e
sec2 x) + etan x
= etan x (x sec2 x + 1).
f 0 (x) = x ·
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