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Part 2
3.5: Derivatives of Trigonometric Functions
Part 2: The Other Basic Functions
MATH 165: Calculus I
Department of Mathematics
Iowa State University
Paul J. Barloon
MATH 165 Section 3.5
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Part 2 The Other Basic Functions
A Brief Review
Recall the derivatives of sin(x) and cos(x):
d
dx
[sin(x)] = cos(x)
d
dx
[cos(x)] = � sin(x)
Using these and our di↵erentiation rules, we get the derivatives ofthe remaining trigonometric functions . . .
MATH 165 Section 3.5
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Part 2 The Other Basic Functions
A Brief Review
Recall the derivatives of sin(x) and cos(x):
d
dx
[sin(x)] = cos(x)
d
dx
[cos(x)] = � sin(x)
Using these and our di↵erentiation rules, we get the derivatives ofthe remaining trigonometric functions . . .
MATH 165 Section 3.5
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Part 2 The Other Basic Functions
Derivative of tan(x)
For f (x) = tan(x), use the definition of the tangent function andthe Quotient Rule:
d
dx
[tan(x)] =d
dx
sin(x)
cos(x)
�
=cos(x)[cos(x)]� sin(x)[� sin(x)]
cos2(x)
=cos2(x) + sin2(x)
cos2(x)
=1
cos2(x)= sec2(x)
MATH 165 Section 3.5
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Part 2 The Other Basic Functions
Derivative of tan(x)
For f (x) = tan(x), use the definition of the tangent function andthe Quotient Rule:
d
dx
[tan(x)] =d
dx
sin(x)
cos(x)
�
=cos(x)[cos(x)]� sin(x)[� sin(x)]
cos2(x)
=cos2(x) + sin2(x)
cos2(x)
=1
cos2(x)= sec2(x)
MATH 165 Section 3.5
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Part 2 The Other Basic Functions
Derivative of tan(x)
For f (x) = tan(x), use the definition of the tangent function andthe Quotient Rule:
d
dx
[tan(x)] =d
dx
sin(x)
cos(x)
�
=cos(x)[cos(x)]� sin(x)[� sin(x)]
cos2(x)
=cos2(x) + sin2(x)
cos2(x)
=1
cos2(x)= sec2(x)
MATH 165 Section 3.5
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Part 2 The Other Basic Functions
Derivative of tan(x)
For f (x) = tan(x), use the definition of the tangent function andthe Quotient Rule:
d
dx
[tan(x)] =d
dx
sin(x)
cos(x)
�
=cos(x)[cos(x)]� sin(x)[� sin(x)]
cos2(x)
=cos2(x) + sin2(x)
cos2(x)
=1
cos2(x)= sec2(x)
MATH 165 Section 3.5
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Part 2 The Other Basic Functions
Derivative of sec(x)
Follow a similar procedure to find the derivative of f (x) = sec(x):
d
dx
[sec(x)] =d
dx
1
cos(x)
�
=cos(x)[0]� (1)[� sin(x)]
cos2(x)
=sin(x)
cos2(x)
=1
cos(x)· sin(x)cos(x)
= sec(x) tan(x)
MATH 165 Section 3.5
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Part 2 The Other Basic Functions
Derivative of sec(x)
Follow a similar procedure to find the derivative of f (x) = sec(x):
d
dx
[sec(x)] =d
dx
1
cos(x)
�
=cos(x)[0]� (1)[� sin(x)]
cos2(x)
=sin(x)
cos2(x)
=1
cos(x)· sin(x)cos(x)
= sec(x) tan(x)
MATH 165 Section 3.5
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Part 2 The Other Basic Functions
Derivative of sec(x)
Follow a similar procedure to find the derivative of f (x) = sec(x):
d
dx
[sec(x)] =d
dx
1
cos(x)
�
=cos(x)[0]� (1)[� sin(x)]
cos2(x)
=sin(x)
cos2(x)
=1
cos(x)· sin(x)cos(x)
= sec(x) tan(x)
MATH 165 Section 3.5
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Part 2 The Other Basic Functions
Derivative of sec(x)
Follow a similar procedure to find the derivative of f (x) = sec(x):
d
dx
[sec(x)] =d
dx
1
cos(x)
�
=cos(x)[0]� (1)[� sin(x)]
cos2(x)
=sin(x)
cos2(x)
=1
cos(x)· sin(x)cos(x)
= sec(x) tan(x)
MATH 165 Section 3.5
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Part 2 The Other Basic Functions
Derivative of cot(x) and csc(x)
Use the same techniques to get the final two trigonometricderivatives:
d
dx
[cot(x)] = � csc2(x)
d
dx
[csc(x)] = � csc(x) cot(x)
MATH 165 Section 3.5
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Part 2 The Other Basic Functions
Summary
Here is the complete list.
d
dx
[sin(x)] = cos(x)
d
dx
[tan(x)] = sec2(x)
d
dx
[sec(x)] = sec(x) tan(x)
d
dx
[cos(x)] = � sin(x)
d
dx
[cot(x)] = � csc2(x)
d
dx
[csc(x)] = � csc(x) cot(x)
MATH 165 Section 3.5
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Part 2 The Other Basic Functions
EXAMPLE 1: Finddr
d✓if r = (4 + sec ✓) sin ✓.
MATH 165 Section 3.5
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Part 2 The Other Basic Functions
Quiz Yourself
Finddp
dq
if p =sin q + cos q
sin q
A)cos q � sin q
cos q
B)� sin q � cos q
sin2 q
C) sec2 q
D) � csc2 q
E) �1
MATH 165 Section 3.5
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Part 2 The Other Basic Functions
Some Good News!
Since all of the trigonometric functions are di↵erentiable, they arecontinuous – at all points where they are defined.
Remember that finding limits of continuous functions – andcomposites of continuous functions – boils down to “plugging in”the value.
So, limits involving trigonometric functions are generallystraightforward . . .
MATH 165 Section 3.5
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Part 2 The Other Basic Functions
Some Good News!
Since all of the trigonometric functions are di↵erentiable, they arecontinuous – at all points where they are defined.
Remember that finding limits of continuous functions – andcomposites of continuous functions – boils down to “plugging in”the value.
So, limits involving trigonometric functions are generallystraightforward . . .
MATH 165 Section 3.5
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Part 2 The Other Basic Functions
Some Good News!
Since all of the trigonometric functions are di↵erentiable, they arecontinuous – at all points where they are defined.
Remember that finding limits of continuous functions – andcomposites of continuous functions – boils down to “plugging in”the value.
So, limits involving trigonometric functions are generallystraightforward . . .
MATH 165 Section 3.5
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Part 2 The Other Basic Functions
EXAMPLE 2: Evaluate the following limit:
limx!0
seche
x + ⇡ tan⇣ ⇡
4 sec x
⌘� 1
i
MATH 165 Section 3.5
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Part 2 The Other Basic Functions
The End
MATH 165 Section 3.5
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Di↵erentiation
Practice With DerivativesPractice Problems
MATH 165: Calculus I
Department of Mathematics
Iowa State University
Paul J. Barloon
MATH 165
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Di↵erentiation Practice Problems
Problem 1: Suppose that a spill is forming acircular oil slick around a leaking tanker.
How fast does the area of the slick change withrespect to its radius when that radius is 1 foot?
A) 1 ft2/ft
B) ⇡ ft2/ft
C) 2⇡ ft2/ft
D) 90⇡ ft2/ft
E) 180⇡ ft2/ft
MATH 165
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Di↵erentiation Practice Problems
Problem 1: Suppose that a spill is forming acircular oil slick around a leaking tanker.
How fast does the area of the slick change withrespect to its radius when that radius is 1 foot?
A) 1 ft2/ft
B) ⇡ ft2/ft
*C) 2⇡ ft2/ft
D) 90⇡ ft2/ft
E) 180⇡ ft2/ft
MATH 165
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Di↵erentiation Practice Problems
Problem 2: Let f (x) =x
4 � x
3 + 5x2 � 3x + 6
x
2 � x + 2.
Find f
0(x).
A) f
0(x) =2x5 � x
4 + 3x3 � 7x + 5
(x2 � x + 2)2
B) f
0(x) =4x3 � 3x2 + 10x � 3
2x � 1
C) f
0(x) = x
2 + 3
D) f
0(x) = 2x
E) f
0(x) = 0
F) f
0(x) DNE
MATH 165
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Di↵erentiation Practice Problems
Problem 2: Let f (x) =x
4 � x
3 + 5x2 � 3x + 6
x
2 � x + 2.
Find f
0(x).
A) f
0(x) =2x5 � x
4 + 3x3 � 7x + 5
(x2 � x + 2)2
B) f
0(x) =4x3 � 3x2 + 10x � 3
2x � 1
C) f
0(x) = x
2 + 3
*D) f
0(x) = 2x
E) f
0(x) = 0
F) f
0(x) DNE
MATH 165
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Di↵erentiation Practice Problems
Problem 3: Suppose that y = x
2g(x) for some
di↵erentiable function g(x).
Find y
0.
A) y
0 = 2x g(x)
B) y
0 = x
2g
0(x)
C) y
0 = 2x g 0(x)
D) y
0 = 2x + g
0(x)
E) y
0 = x (x g(x) + 2 g 0(x))
F) y
0 = x (x g 0(x) + 2 g(x))
MATH 165
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Di↵erentiation Practice Problems
Problem 3: Suppose that y = x
2g(x) for some
di↵erentiable function g(x).
Find y
0.
A) y
0 = 2x g(x)
B) y
0 = x
2g
0(x)
C) y
0 = 2x g 0(x)
D) y
0 = 2x + g
0(x)
E) y
0 = x (x g(x) + 2 g 0(x))
*F) y
0 = x (x g 0(x) + 2 g(x))
MATH 165
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Di↵erentiation Practice Problems
Problem 4: Suppose that y =g(x)
x
2for some
di↵erentiable function g(x).
Find y
0.
A) y
0 =g
0(x)
2xD) y
0 =x g
0(x)� 2 g(x)
x
B) y
0 =2x
g
0(x)E) y
0 =2 g(x)� x g
0(x)
x
3
C) y
0 =2 g(x)� x g
0(x)
x
F) y
0 =x g
0(x)� 2 g(x)
x
3
MATH 165
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Di↵erentiation Practice Problems
Problem 4: Suppose that y =g(x)
x
2for some
di↵erentiable function g(x).
Find y
0.
A) y
0 =g
0(x)
2xD) y
0 =x g
0(x)� 2 g(x)
x
B) y
0 =2x
g
0(x)E) y
0 =2 g(x)� x g
0(x)
x
3
C) y
0 =2 g(x)� x g
0(x)
x
*F) y
0 =x g
0(x)� 2 g(x)
x
3
MATH 165
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Di↵erentiation Practice Problems
Problem 5a: Letf (x) = x
20 = 10x17�⇡5x
12+ 23x
9�11.1x7+3x2�5.
Find f
(21)(x).
A) 20x19 � 170x16 � 60⇡4x
11 + 6x8 � 77.7x6 + 6x
B) 20x19 � 170x16 � 12⇡5x
11 + 6x8 � 77.7x6 + 6x
C) 20x � 10
D) 20x
E) (20!)x�1
F) 0
MATH 165
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Di↵erentiation Practice Problems
Problem 5a: Letf (x) = x
20 = 10x17�⇡5x
12+ 23x
9�11.1x7+3x2�5.
Find f
(21)(x).
A) 20x19 � 170x16 � 60⇡4x
11 + 6x8 � 77.7x6 + 6x
B) 20x19 � 170x16 � 12⇡5x
11 + 6x8 � 77.7x6 + 6x
C) 20x � 10
D) 20x
E) (20!)x�1
*F) 0
MATH 165
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Di↵erentiation Practice Problems
Problem 5b: Letf (x) = x
20 = 10x17�⇡5x
12+ 23x
9�11.1x7+3x2�5.
Find f
(20)(x).
A) 20x
B) (20!)x
C) 20x�1
D) 20!
E) 20
F) 0
MATH 165
![Page 39: 3.5: Derivatives of Trigonometric Functionsorion.math.iastate.edu/miriamc/M165F18/3-52_Notes165_F18.pdfPart 2 The Other Basic Functions ABriefReview Recall the derivatives of sin(x)](https://reader034.vdocuments.us/reader034/viewer/2022050601/5fa83e5e7161bb57140e53b7/html5/thumbnails/39.jpg)
Di↵erentiation Practice Problems
Problem 5b: Letf (x) = x
20 = 10x17�⇡5x
12+ 23x
9�11.1x7+3x2�5.
Find f
(20)(x).
A) 20x
B) (20!)x
C) 20x�1
*D) 20!
E) 20
F) 0
MATH 165
![Page 40: 3.5: Derivatives of Trigonometric Functionsorion.math.iastate.edu/miriamc/M165F18/3-52_Notes165_F18.pdfPart 2 The Other Basic Functions ABriefReview Recall the derivatives of sin(x)](https://reader034.vdocuments.us/reader034/viewer/2022050601/5fa83e5e7161bb57140e53b7/html5/thumbnails/40.jpg)
Di↵erentiation Practice Problems
The End
MATH 165
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MATH 165 44–49 Warm-Up Question – Sep. 19, 2018
Use the Quotient Rule to find the derivative
of the following function:
f (x) =
1
cos(x)
A) f
0(x) = 0 D) f
0(x) = sec(x) tan(x)
B) f
0(x) = � 1
sin(x)
E) f
0(x) = � sin(x)
cos
2(x)
C) f
0(x) =
sin(x)
cos
2(x)
F) f
0(x) = � sec(x) tan(x)
![Page 42: 3.5: Derivatives of Trigonometric Functionsorion.math.iastate.edu/miriamc/M165F18/3-52_Notes165_F18.pdfPart 2 The Other Basic Functions ABriefReview Recall the derivatives of sin(x)](https://reader034.vdocuments.us/reader034/viewer/2022050601/5fa83e5e7161bb57140e53b7/html5/thumbnails/42.jpg)
![Page 43: 3.5: Derivatives of Trigonometric Functionsorion.math.iastate.edu/miriamc/M165F18/3-52_Notes165_F18.pdfPart 2 The Other Basic Functions ABriefReview Recall the derivatives of sin(x)](https://reader034.vdocuments.us/reader034/viewer/2022050601/5fa83e5e7161bb57140e53b7/html5/thumbnails/43.jpg)
MATH 165 44–49 Warm-Up Question – Sep. 19, 2018
Use the Quotient Rule to find the derivative
of the following function:
f (x) =
1
cos(x)
A) f
0(x) = 0 *D) f
0(x) = sec(x) tan(x)
B) f
0(x) = � 1
sin(x)
E) f
0(x) = � sin(x)
cos
2(x)
*C) f
0(x) =
sin(x)
cos
2(x)
F) f
0(x) = � sec(x) tan(x)