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AP Calculus BC
Unit 2: Differentiation: Definition and Fundamental Properties
2.10 Finding the Derivatives of Trigonometric Functions
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Cards (9)
The derivative of sin(x) is
cos(x)
The derivative of cos(x) is
-sin(x)
The derivative of tan(x) is
sec
2
x
\sec^{2}x
sec
2
x
The derivative of sec(x) is
s
e
c
x
t
a
n
x
secx tanx
sec
x
t
an
x
The derivative of cot(x) is
−
csc
2
x
- \csc^{2}x
−
csc
2
x
The derivative of csc(x) is
−
c
s
c
x
c
o
t
x
- cscx cotx
−
csc
x
co
t
x
The derivatives of trigonometric functions can be derived using the
product rule
.
The quotient rule is used to find the derivative of
tan(x)
.
Steps to derive the derivative of tan(x) using the quotient rule.
1️⃣ Write
t
a
n
(
x
)
tan(x)
t
an
(
x
)
as
sin
(
x
)
cos
(
x
)
\frac{\sin(x)}{\cos(x)}
c
o
s
(
x
)
s
i
n
(
x
)
2️⃣ Apply the quotient rule
3️⃣ Simplify the derivative using
cos
2
(
x
)
+
\cos^{2}(x) +
cos
2
(
x
)
+
sin
2
(
x
)
=
\sin^{2}(x) =
sin
2
(
x
)
=
1
1
1
4️⃣ Rewrite the result as
sec
2
(
x
)
\sec^{2}(x)
sec
2
(
x
)