Deck 33: Derivatives of Trigonometric, Logarithmic, and Exponential Functions

Question 1

Find the derivative: $y=3 \cos^{4} x$

Answer 1

$\frac{d y}{d x}=-12 \cos^{3} x \sin x$

Question 2

Find the derivative: $y=\sin^{3} x \cos x$

Answer 2

$\frac{d y}{d x}=3 \sin^{2} x \cos^{2} x-\sin^{4} x$

Question 3

Find the derivative: $y=\sin 3 x \cos 2 x$

Answer 3

$3 \cos 3 x \cos 2 x-2 \sin 3 x \sin 2 x$

Question 4

Find the second derivative of the function $y=\sin x \cos x$ .

Question 5

Find $\frac{d y}{d x}$ for the implicit function $x=\sin (y-x)$ .

Question 6

Find the slope of the tangent line at the given value of $x . y=\cos 2 x$ at $x=2$ rad, to two decimal places.

Question 7

Find the maximum, minimum, and inflection points of the function $y=\sin x \cos x$ between $x=0$ and $x=2 \pi$ .

Question 8

Sketch the function $y=\cos^{2} x-x$ . Calculate a root to three decimal places by Newton's method. If there is more than one root, find only the smallest positive root.

Question 9

Find the derivative: $y=x \cos \pi x$

Question 10

Find the derivative: $y=\sin^{2}(x-\pi)$

Question 11

Find the derivative: $y=\sin 2 x \cos 2 x$

Question 12

Find $\frac{d y}{d x}$ for the implicit function: $x^{2}+y^{2}=\sin x y$

Question 13

Find the slope of the tangent at the given value of $x: y=\sin x \cos^{2} x$ at $x=\pi$

Question 14

Differentiate implicitly: $x y^{2}=x \cos y+y \sin x$

Question 15

Use Newton's method to find a root of $y=x \sin x^{2}$ on [2,3], to three decimal places.

Question 16

Find the second derivative of $y=\sin^{2} x$ .

Question 17

Find the derivative of $y=5 x^{4}-3 \cos 6 x$ .

Question 18

Find the second derivative of $y=4 \sin \left(2 x-\frac{\pi}{4}\right)$ .

Question 19

Find the slope of the tangent to the curve $y=x^{2} \sin x^{2}$ at $x=0.5$ .

Question 20

Find $\frac{d y}{d x}$ for $\sin (x+y)=x y$ .

Question 21

Find $f^{\prime}(2)$ for $f(x)=5 \cos \frac{x}{3}-x^{3}$ . Express your answer to three significant digits.

Question 22

Find the derivative of $y=\frac{\cos x}{x^{2}}$ .

Question 23

Find the derivative of $y=2 x^{3} \sin 2 x$ .

Question 24

If $f(x)=\cos (1-\pi x)$ , find $f^{\prime \prime}\left(\frac{\pi}{2}\right)$ . Express your answer to three significant digits.

Question 25

Differentiate: $y=\sqrt{\tan 2 \theta}$

Question 26

Differentiate: $y=\sin 3 x \sec 5 x$

Question 27

Differentiate: $y=\csc^{n} a x$

Question 28

The potential energy of a spring with a constant of $k$ , a maximum amplitude of $A$ , and an angular velocity of $\omega$ is given by $P=\frac{1}{2} k A^{2} \cos^{2} \omega t$
(joules) where $t$ is in seconds. Find the rate of change in the potential energy of a mass on a spring with a constant of $1.2 \mathrm{~N} / \mathrm{m}$ , a maximum amplitude of $0.53 \mathrm{~m}$ , and an angular velocity of $0.031 \mathrm{rad} / \mathrm{s}$ , when $t=5.8 \mathrm{~s}$ .

Question 29

A kite rises straight up to $12.0 \mathrm{~m}$ , where it stops rising and then starts moving parallel to the ground at $2.10 \mathrm{~m} / \mathrm{s}$ . How fast is the angle between the kite string and the ground changing, in rad/s, $4.00 \mathrm{~s}$ after the kite starts to move horizontally? Assume the string is perfectly straight.

Question 30

The acceleration of a block of mass $m$ being pulled by a constant force $F$ which is applied at an angle above the horizontal is given by $a=\frac{F \cos \theta-f m g+f F \sin \theta}{m}$ where $f$ is the coefficient of friction between the block and the surface and $g$ is the acceleration due to gravity. Find the angle that maximizes the block's acceleration when $f=0.52$ .

Question 31

Find the derivative: $y=\tan 3 x \sec x$

Question 32

Find the derivative: $y=x^{2} \sec^{2} x$

Question 33

Find the derivative: $y=5 \tan 2 \pi x-3 \cos 4 \pi x$

Question 34

Find the second derivative: $y=2 \tan 5 x$

Question 35

Find $\frac{d y}{d x}$ for the implicit function: $4 \sec y=x y-2$

Question 36

Find the equation of the tangent to the curve $y=\sec x \tan^{2} x+x$ at $x=\pi$ .

Question 37

An airplane is flying horizontally at an altitude of $5 \mathrm{~km}$ and with speed of $600 \mathrm{~km} / \mathrm{h}$ . It passes directly over an observer on the ground. Determine how fast the angle between the airplane and the observer is increasing, in rads/h, 1 minute after the airplane passed the observer.

Question 38

Use Newton's method to find a root of $y{=x+\tan x}$ on [4,5], to three decimal places.

Question 39

Find the derivative of $y=\csc x^{2}$ .

Question 40

Find the derivative of $y=\frac{2 \sin 3 x}{3 \cot 2 x}$ .

Question 41

Find the derivative of $y=\frac{\tan x^{2}}{3}$ .

Question 42

Find the value of the derivative of $y=4 \tan x^{2} at x=\frac{1}{4}$ .

Question 43

Find $\frac{d y}{d x}$ of $3 x^{2}-2 \csc (y-x)=1$ .

Question 44

The blade of a reciprocating saw has a displacement (in $\mathrm{m}$ ) of $y=2.5 \sin 8 \pi t-1.5 \cos 4 \pi t$ . Find the velocity of the blade at 2.0 s.

Question 45

Find the second derivative: $y=\sec 2 x$

Question 46

Find the derivative:
$r=\frac{\cot \theta}{\theta}$

Question 47

What is the maximum length of a board that you can fit around a $90^{\circ}$ corner if one hallway is $2.8 \mathrm{~m}$ wide and the other hallway is $1.4 \mathrm{~m}$ wide? (Round to 4 significant digits.)

Question 48

Differentiate: $y = tan^{-1}\sqrt{1+x^2}$

Question 49

Differentiate: $y=5 x \tan^{-1} \frac{1}{x}$

Question 50

Differentiate: $y=\cot^{-1} \frac{3 x}{\sqrt{9 x^{2}-1}}$

Question 51

Find the slope of the tangent to the curve: $y=\sqrt{\operatorname{arc cot sin} x}$ at $x=1$ , to three decimal places

Question 52

Find the derivative: $y=x \arctan 2 x$

Question 53

Find the derivative: $y=\sqrt{x} \operatorname{arccsc} x$

Question 54

Find the equations of the tangents to the curve $y=\tan^{-1} x$ having a slope of $\frac{1}{2}$ .

Question 55

Find the slope of the tangent to the curve $y=x-\arcsin x^{2}$ at $x=0.5$ .

Question 56

Find the derivative of $y=5 \csc^{-1} x^{3}$ .

Question 57

Find the second derivative of $y=3 \tan^{-1} 2 x$ .

Question 58

Find the derivative of $y=\frac{1}{2} \sin^{-1} \frac{4}{x}$ .

Question 59

Find the slope of the tangent to $y=2 \cos^{-1} \sqrt{x}$ at $x=0.5$ .

Question 60

Find the derivative of $y=x \sec^{-1} x$ .

Question 61

Find the derivative of $y=8 \sec^{-1} 2 x$ .

Question 62

Find the derivative of $y=\frac{x}{2} \tan^{-1} \frac{x}{2}$ .

Question 63

Determine the derivative: $y=\log _{a}(x-1)^{3}$

Question 64

Determine the derivative: $y = ln x^2$

Question 65

Determine the derivative: $y=\ln \sqrt{\frac{x^{2}-1}{x^{2}+1}}$

Question 66

Determine the derivative: $y=\ln \cos x$

Question 67

Determine $\frac{d y}{d x}: 2 \ln y+\ln \left(x^{2}+4\right)=1$

Question 68

Take the logarithm of both sides and then differentiate to determine $\frac{d y}{d x}: y=(x-2)^{x}$

Question 69

Find the slope of the tangent at the given value of $x: y=\log x^{2}$ at $x=5$ , to three decimal places

Question 70

Find the angle of intersection of $y=\ln x^{2}$ and $y=\ln (6-x)$ at $x=2$ .

Question 71

Use Newton's method to find the smallest positive root of the equation $y=(\sin x) \ln (\sin x)$ .

Question 72

Differentiate: $y=2 \log _{b}\left(x^{2}+3\right)$

Question 73

Differentiate: $y=x \log \sqrt{10 x+1}$

Question 74

Differentiate: $y=\ln \sqrt{x^{2}+4 x+1}$

Question 75

Differentiate: $y=\ln \left(\sin \left(x^{2}+1\right)\right)$

Question 76

Differentiate implicitly: $x^{2}+y^{2}=x \ln y$

Question 77

Determine the derivative of $y=\ln \left(x^{2}+7\right)$ .

Question 78

Find $\frac{d y}{d x}$ for $y=\ln (x \sec x)$ .

Question 79

Find $\frac{d y}{d x}$ for $y=\ln \left(x^{2} y\right)$ .

Question 80

Find the derivative of $y=8 \log _{2}(1-4 x)$ .