Trig Functions and the Chain Rule - Texas A&M University
Lecture for Week 5 (Secs. 3.4?5)
Trig Functions and the Chain Rule
1
The important differentiation formulas for trigonometric functions are
d dx
sin
x
=
cos
x,
d dx
cos
x
=
-
sin
x.
Memorize them! To evaluate any other trig deriva-
tive, you just combine these with the product (and
quotient) rule and chain rule and the definitions of
the other trig functions, of which the most impor-
tant is
tan x = sin x . cos x
2
Prove that
Exercise 3.4.19
d dx
cot
x
=
-
csc2
x.
Exercise 3.4.23
Find the derivative of y = csc x cot x.
3
What
is
d dx
cot
x
?
Well,
the
definition
of
the
cotangent is
cot x
=
cos x sin x
.
So, by the quotient rule, its derivative is
sin x(- sin x) - cos x(cos x)
sin2 x
=
-
1 sin2
x
- csc2
x
(since sin2 x + cos2 x = 1).
4
To differentiate csc x cot x use the product rule:
dy dx
=
d csc x dx
cot x + csc x
d cot x dx
.
The second derivative is the one we just calculated, and the other one is found similarly (Ex. 3.4.17):
d dx
csc
x
=
-
csc
x
cot
x.
5
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