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Unit 2: Differentiation: Definition and Fundamental Properties
2.7 Derivatives of Basic Functions
Derivatives of trigonometric functions:
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Cards (182)
What are the six basic trigonometric functions?
sin x, cos x, tan x, csc x, sec x, cot x
The derivative of cos x using the limit definition is
-sin x
The chain rule can be applied to differentiate composite
trigonometric
functions.
True
The notation dy/dx represents the rate of change of
y
with respect to x.
The notation dy/dx represents the rate of change of y with respect to
x
Steps to find the derivative of sin x using the limit definition
1️⃣ Substitute f(x) = sin x into the limit definition
2️⃣ Use the angle sum identity sin(x + h) = sin x cos h + cos x sin h
3️⃣ Separate the limit into two terms
4️⃣ Apply the known limits
lim
h
→
0
cos
h
−
1
h
=
\lim_{h \to 0} \frac{\cos h - 1}{h} =
lim
h
→
0
h
c
o
s
h
−
1
=
0
0
0
and
lim
h
→
0
sin
h
h
=
\lim_{h \to 0} \frac{\sin h}{h} =
lim
h
→
0
h
s
i
n
h
=
1
1
1
5️⃣ Simplify to obtain f'(x) = cos x
The angle sum identity for cos(x + h) is cos x cos h - sin x sin
h
Steps to find the derivative of cos x using the limit definition
1️⃣ Substitute f(x) = cos x into the limit definition
2️⃣ Use the angle sum identity cos(x + h) = cos x cos h - sin x sin h
3️⃣ Separate the limit into two terms
4️⃣ Apply the known limits
lim
h
→
0
cos
h
−
1
h
=
\lim_{h \to 0} \frac{\cos h - 1}{h} =
lim
h
→
0
h
c
o
s
h
−
1
=
0
0
0
and
lim
h
→
0
sin
h
h
=
\lim_{h \to 0} \frac{\sin h}{h} =
lim
h
→
0
h
s
i
n
h
=
1
1
1
5️⃣ Simplify to obtain f'(x) = -sin x
Match the trigonometric function with its ratio in a right triangle:
sin x ↔️ Opposite / Hypotenuse
cos x ↔️ Adjacent / Hypotenuse
tan x ↔️ Opposite / Adjacent
What does dy/dx represent in calculus notation?
Instantaneous rate of change
What is the formula for f'(x) using the limit definition of the derivative?
f
′
(
x
)
=
f'(x) =
f
′
(
x
)
=
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
lim
h
→
0
h
f
(
x
+
h
)
−
f
(
x
)
Derivatives are written using two common notations:
dy/dx
and f'(x)
The derivative of tan x can be found using the
quotient rule
.
True
Match the trigonometric function with its ratio:
sin x ↔️ Opposite / Hypotenuse
cos x ↔️ Adjacent / Hypotenuse
tan x ↔️ Opposite / Adjacent
csc x ↔️ Hypotenuse / Opposite
sec x ↔️ Hypotenuse / Adjacent
cot x ↔️ Adjacent / Opposite
Steps to find the derivative of sin x using the limit definition
1️⃣ Substitute f(x) = sin x
2️⃣ Use the angle sum identity sin(x + h) = sin x cos h + cos x sin h
3️⃣ Separate the limit
4️⃣ Apply known limits
5️⃣ Simplify the expression
What does f'(x) represent in calculus?
Derivative of f(x)
What is the derivative of sin x using the limit definition?
cos x
What is the derivative of cos x using the limit definition?
-sin x
The derivative of sin x is
cos
x
Derivatives represent the
instantaneous
rate of change of a function
True
If y = sin x, then dy/dx =
cos x
If f(x) = x<sup>2</sup>, then f'(x) =
2x
What do derivatives represent in calculus?
Instantaneous rate of change
f'(x) is the
derivative
of the function f(x).
True
What is the first step in finding the derivative of sin x using the limit definition?
Apply the limit definition
What is the value of
lim
h
→
0
cos
h
−
1
h
\lim_{h \to 0} \frac{\cos h - 1}{h}
lim
h
→
0
h
c
o
s
h
−
1
?
0
What is the derivative of cos x?
-sin x
What is the derivative of cos x?
-sin x
What trigonometric identity is used to simplify the derivative of tan x?
cos² x + sin² x = 1
The quotient rule is necessary to find the derivative of tan x
True
The trigonometric identity cos² x +
sin²
x equals 1
True
Steps to find the derivative of tan x using the quotient rule
1️⃣ Express tan x as a quotient:
tan
x
=
\tan x =
tan
x
=
sin
x
cos
x
\frac{\sin x}{\cos x}
c
o
s
x
s
i
n
x
2️⃣ Identify u and v:
u
=
u =
u
=
sin
x
,
v
=
\sin x, v =
sin
x
,
v
=
cos
x
\cos x
cos
x
3️⃣ Find u' and v':
u
′
=
u' =
u
′
=
cos
x
,
v
′
=
\cos x, v' =
cos
x
,
v
′
=
−
sin
x
- \sin x
−
sin
x
4️⃣ Apply the quotient rule:
d
d
x
(
tan
x
)
=
\frac{d}{dx}(\tan x) =
d
x
d
(
tan
x
)
=
cos
x
⋅
cos
x
−
sin
x
⋅
(
−
sin
x
)
cos
2
x
\frac{\cos x \cdot \cos x - \sin x \cdot ( - \sin x)}{\cos^{2} x}
c
o
s
2
x
c
o
s
x
⋅
c
o
s
x
−
s
i
n
x
⋅
(
−
s
i
n
x
)
5️⃣ Simplify:
d
d
x
(
tan
x
)
=
\frac{d}{dx}(\tan x) =
d
x
d
(
tan
x
)
=
sec
2
x
\sec^{2} x
sec
2
x
The notation dy/dx represents the rate of change of y with respect to
x
The derivative of cos x is
-sin x
What is the value of
lim
h
→
0
sin
h
h
\lim_{h \to 0} \frac{\sin h}{h}
lim
h
→
0
h
s
i
n
h
?
1
The trigonometric identity \sin(x + h) = \sin x \cos h + \cos x \sin h</latex> is used to differentiate
sin
x
\sin x
sin
x
using the limit definition.
True
What is the formula for the limit definition of the derivative?
f
′
(
x
)
=
f'(x) =
f
′
(
x
)
=
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
lim
h
→
0
h
f
(
x
+
h
)
−
f
(
x
)
Steps to differentiate
cos
x
\cos x
cos
x
using the limit definition.
1️⃣ Apply the limit definition:
f
′
(
x
)
=
f'(x) =
f
′
(
x
)
=
lim
h
→
0
cos
(
x
+
h
)
−
cos
x
h
\lim_{h \to 0} \frac{\cos(x + h) - \cos x}{h}
lim
h
→
0
h
c
o
s
(
x
+
h
)
−
c
o
s
x
2️⃣ Use the angle sum identity:
cos
(
x
+
h
)
=
\cos(x + h) =
cos
(
x
+
h
)
=
cos
x
cos
h
−
sin
x
sin
h
\cos x \cos h - \sin x \sin h
cos
x
cos
h
−
sin
x
sin
h
3️⃣ Separate the limit:
lim
h
→
0
(
cos
x
cos
h
−
1
h
−
sin
x
sin
h
h
)
\lim_{h \to 0} \left( \cos x \frac{\cos h - 1}{h} - \sin x \frac{\sin h}{h} \right)
lim
h
→
0
(
cos
x
h
c
o
s
h
−
1
−
sin
x
h
s
i
n
h
)
4️⃣ Apply known limits:
lim
h
→
0
cos
h
−
1
h
=
\lim_{h \to 0} \frac{\cos h - 1}{h} =
lim
h
→
0
h
c
o
s
h
−
1
=
0
,
lim
h
→
0
sin
h
h
=
0, \lim_{h \to 0} \frac{\sin h}{h} =
0
,
lim
h
→
0
h
s
i
n
h
=
1
1
1
5️⃣ Simplify to final derivative:
f
′
(
x
)
=
f'(x) =
f
′
(
x
)
=
−
sin
x
- \sin x
−
sin
x
To differentiate
tan
x
\tan x
tan
x
using the quotient rule, it is expressed as \frac{\sin x}{\cos x}</latex>.
True
If y = sin x, then dy/dx = cos x.
True
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