Deck 3: Applications of Differentiation

Question 1

Find the derivative of the function. $f(x)$ = sinh 4x

A) -4 cosh 4x
B) 4 sinh 4x
C) -sinh 4x
D) 4 cosh 4x

Answer 1

4 cosh 4x

Question 2

Find the given integral. $\int \cosh (9 x+5) d x$

A) -sinh (9x + 5)+ C
B) 9sinh (9x + 5) + C
C) sinh (9x + 5)+ C
D) $\frac{1}{9}$ sinh (9x + 5) + C

Answer 2

$\frac{1}{9}$ sinh (9x + 5) + C

Question 3

Find the derivative of .

Answer 3

Question 4

A telephone line hangs between two poles at 12 m apart in the shape of the catenary $y=30 \cosh \left(\frac{x}{30}\right)-35$ , where x and y are measured in meters. Find the slope of this curve where it meets the right pole.

A) $\frac{\sinh 5}{6}$
B) $\sinh \left(\frac{5}{6}\right)$
C) $\sinh \left(\frac{1}{6}\right)$
D) $\sinh 6$
E) $\sinh \left(\frac{1}{5}\right)$

Question 5

Two sides of a triangle are 2 m and 3 m in length and the angle between them is increasing at a rate of $0.07$ rad/s. Find the rate at which the area of the triangle is increasing when the
Angle between the sides of fixed length is ( $\frac{\pi}{3}$ )

A) $-0.955 \mathrm{~m}^{2} / \mathrm{s}$
B) $1.145 \mathrm{~m}^{2} / \mathrm{s}$
C) $-1.955 \mathrm{~m}^{2} / \mathrm{s}$
D) $0.105 \mathrm{~m}^{2} / \mathrm{s}$
E) $5.045 \mathrm{~m}^{2} / \mathrm{s}$

Question 6

Find the derivative of the function. = sinh ^-1 6x

Question 7

Determine the values of x for which the given linear approximation is accurate to within 0.07 at a = 0. $\tan x \approx X$

A) $-0.19<$$x<0.28$
B) $-0.57<$$x<0.57$
C) $0.06<$$x<0.68$
D) $-1.04<$$x<1.55$
E) $-0.71<$$x<0.48$

Question 8

Use the linear approximation of the function $f(x)=\sqrt{9-x}$ at $a=0$ to approximate the number $\sqrt{9.08}$ .

A) $7.4445$
B) $2.2556$
C) $3.0133$
D) $7.4556$
E) $0.1556$

Question 9

A turkey is removed from the oven when its temperature reaches $175^{\circ} \mathrm{F}$ and is placed on a table in a room where the temperature is $70^{\circ} \mathrm{F}$ . After 10 minutes the temperature of the turkey is $160^{\circ} \mathrm{F}$ and after 20 minutes it is $150^{\circ} \mathrm{F}$ . Use a linear approximation to predict the temperature of the turkey after $30$ minutes.

A) $160$
B) $36$
C) $134$
D) $135$
E) $140$

Question 10

The top of a ladder slides down a vertical wall at a rate of $0.1$ m/s . At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s . How long is the ladder?

A) $7 \mathrm{~m}$
B) $2.3 \mathrm{~m}$
C) $2 \mathrm{~m}$
D) $2.8 \mathrm{~m}$
E) None of these

Question 11

Find the derivative of the function. $f(t)$ = cosh² (6t² + 3)

A) 24t sinh (6t² + 3)
B) 24t cosh (6t² + 3) sinh (6t² + 3)
C) 12t sinh (6t² + 3)
D) 12t cosh (6t² + 3) sinh (6t² + 3)

Question 12

Find the value of the expression accurate to four decimal places. sinh 4

A) 55.5798
B) 15.145
C) 27.2899
D) 29.3082

Question 13

Evaluate

Question 14

Gravel is being dumped from a conveyor belt at a rate of 32 ft/min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high? Round the result to the nearest hundredth.

A) $0.41 ~\mathrm{ft} / \min$
B) $1.24 ~\mathrm{ft} / \min$
C) $0.14 ~\mathrm{ft} / \min$
D) $0.27 ~\mathrm{ft} / \min$
E) $0.6 ~\mathrm{ft} / \min$

Question 15

If two resistors with resistances $R_{1}$ and $R_{2}$ are connected in parallel, as in the figure, then the total resistance $R$ measured in ohms ( $\Omega$ ), is given by $\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}$ . If $R_{1}$ and $R_{2}$ are increasing at rates of $0.1 \Omega / s$ and $0.2 \Omega / s$ respectively, how fast is $R$ changing when $R_{1}=75$ and $R_{2}=100$ ?
Round the result to the nearest thousandth.

A) $0.1596 ~\Omega / \mathrm{s}$
B) $1.1974 ~\Omega / \mathrm{s}$
C) $0.1454 ~\Omega / \mathrm{s}$
D) $0.0694 ~\Omega / \mathrm{s}$
E) $0.1688 ~\Omega / \mathrm{s}$

Question 16

Find the derivative of the function. y = $\sqrt{36 x^{2}-1}-6 \cosh ^{-1} 6 x$

A) $\frac{36}{\sqrt{36 x^{2}-1}}$
B) $\frac{36(x-1)}{\sqrt{36 x^{2}-1}}$
C) $\frac{6(x-1)}{\sqrt{36 x^{2}-1}}$
D) $\frac{6(x-1)}{\sqrt{6 x^{2}-1}}$

Question 17

Use differentials to estimate the amount of paint needed to apply a coat of paint $0.0018$ cm thick to a hemispherical dome with diameter $50$ m.

A) $2.25 \pi$
B) $2.28 \pi$
C) $3.82 \pi$
D) $4.11 \pi$
E) $2.52 \pi$

Question 18

A plane flying horizontally at an altitude of 1 mi and a speed of $520$ mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station.

A) $\approx 455 ~\mathrm{mi} / \mathrm{h}$
B) $\approx 570~ \mathrm{mi} / \mathrm{h}$
C) $\approx 495~ \mathrm{mi} / \mathrm{h}$
D) $\approx 670~ \mathrm{mi} / \mathrm{h}$
E) $\approx 450~ \mathrm{mi} / \mathrm{h}$

Question 19

The circumference of a sphere was measured to be cm with a possible error of cm. Use differentials to estimate the maximum error in the calculated volume.

Question 20

Two cars start moving from the same point. One travels south at $50$ mi/h and the other travels west at $40$ mi/h. At what rate is the distance between the cars increasing 2 hours later? Round the result to the nearest hundredth.

A) $55.42 ~\mathrm{mi} / \mathrm{h}$
B) $76.43 ~\mathrm{mi} / \mathrm{h}$
C) $81.38 ~\mathrm{mi} / \mathrm{h}$
D) $65.49 ~\mathrm{mi} / \mathrm{h}$
E) $64.03~ \mathrm{mi} / \mathrm{h}$

Question 21

The volume of a cube is increasing at a rate of . How fast is the surface area increasing when the length of an edge is .

Question 22

The quantity Q of charge in coulombs C that has passed through a point in a wire up to time t (measured in seconds) is given by $Q(t)=t^{3}-3 t^{2}+4 t+3$ . Find the current when $t=3 s$ .

A) 15
B) 24
C) 26
D) 18
E) 13

Question 23

The top of a ladder leaning against a wall is 8 ft above the ground. The slope of the ladder with respect to the ground is -4. What is the length of the ladder?

Question 24

Water flows from a tank of constant cross-sectional area 50 $\mathrm{ft}^{2}$ through an orifice of constant cross-sectional area $\frac{1}{4}$ $\mathrm{ft}^{2}$ located at the bottom of the tank. Initially, the height of the water in the tank was 20 ft, and t sec later it was given by the equation $2 \sqrt{h}+\frac{1}{25} t-2 \sqrt{20}=0 \quad \quad \quad 0 \leq t \leq 50 \sqrt{20}$ How fast was the height of the water decreasing when its height was 2 ft?

A) $100 \sqrt{5}-50 \sqrt{2}$ ft/sec
B) $100 \sqrt{5}-50 \sqrt{2}$ ft/sec.
C) $\frac{2}{25}$ ft/sec
D) $\frac{\sqrt{2}}{25}$ ft/sec

Question 25

The altitude of a triangle is increasing at a rate of while the area of the triangle is increasing at a rate of . At what rate is the base of the triangle changing when the altitude is 10 cm and the area is .

Question 26

Suppose the daily total cost (in dollars) of manufacturing x televisions is $C(x)=0.0004 x^{3}-0.08 x^{2}+160 x+7000$ What is the marginal cost when x = 300? What is the actual cost incurred in manufacturing the 301st television?

A) $195.33, $195.42
B) $220.00, $220.28
C) $195.33, $195.98
D) $220.00, $220.73

Question 27

Find the average rate of change of the area of a circle with respect to its radius r as r changes from 5 to 6 .

A) $11 \pi$
B) $36 \pi$
C) $8 \pi$
D) $6 \pi$
E) $12 \pi$

Question 28

In an adiabatic process (one in which no heat transfer takes place), the pressure P and volume V of an ideal gas such as oxygen satisfy the equation $p^{5} V^{7}=C$ , where C is a constant. Suppose that at a certain instant of time, the volume of the gas is 2L, the pressure is 100 kPa, and the pressure is decreasing at the rate of 5 kPa/sec. Find the rate at which the volume is changing.

A) 14 L/sec
B) $C-$ 14 L/sec
C) $C-$ $\frac{1}{14}$ L/sec
D) $\frac{1}{14}$ L/sec

Question 29

The parents of a child wish to establish a trust fund for the child's college education. If they need an estimated $90,000 5 years from now and they are able to invest the money at 5.5% compounded continuously in the interim, how much should they set aside in trust now?

A) $68,361.49
B) $17,061.61
C) $17,036.73
D) $68,862.09

Question 30

If a snowball melts so that its surface area decreases at a rate of , find the rate at which the diameter decreases when the diameter is cm.

Question 31

A water trough is 20 m long and a cross-section has the shape of an isosceles trapezoid that is 20 cm wide at the bottom, 60 cm wide at the top, and has height 50 cm. If the trough is being filled with water at the rate of , how fast is the water level rising when the water is cm deep? Round the result to the nearest hundredth.

Question 32

A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of ft/s. At what rate is his distance from second base decreasing when he is halfway to first base? Round the result to the nearest hundredth.

Question 33

If two resistors with resistances $R_{1}$ and $R_{2}$ are connected in parallel, as in the figure, then the total resistance $R$ measured in ohms ( $\Omega$ ), is given by $\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}$ . If $R_{1}$ and $R_{2}$ are increasing at rates of $0.1 ~\Omega / s$ and $0.2 ~\Omega / s$ respectively, how fast is $R$ changing when $R_{1}=75$ and $R_{2}=100$ ?
Round your answer to the nearest thousandth.

A) $0.1596 ~\Omega / \mathrm{s}$
B) $0.1454 ~\Omega / \mathrm{s}$
C) $1.1974 ~\Omega / \mathrm{s}$
D) $0.1688 ~\Omega / \mathrm{s}$
E) $0.0694 ~\Omega / \mathrm{s}$

Question 34

Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley (see the figure below). The point Q is on the floor 12 ft directly beneath and between the carts. Cart A is being pulled away from Q at a speed of ft/s. How fast is cart B moving toward Q at the instant when cart A is 5 ft from Q?

Question 35

The equation of motion is given for a particle, where s is in meters and t is in seconds. Find the acceleration after $2.5$ seconds. $s=\sin 2 \pi t$

A) $0 \mathrm{~m} / \mathrm{s}^{2}$
B) $-15.625 ~\pi^{2} \mathrm{~m} / \mathrm{s}^{2}$
C) $-6.25 ~\pi \mathrm{m} / \mathrm{s}^{2}$
D) $15.625 \pi^{2} \mathrm{~m} / \mathrm{s}^{2}$
E) $6.25 \pi \mathrm{m} / \mathrm{s}^{2}$

Question 36

Find the rate of change of y with respect of x at the indicated value of x. t = csc x - 18 cos x; $x=\frac{\pi}{6}$

A) $9+\frac{2 \sqrt{3}}{3}$
B) $9-\frac{2 \sqrt{3}}{3}$
C) $9-2 \sqrt{3}$
D) $9+2 \sqrt{3}$

Question 37

Find an equation of the tangent line to the curve $90\left(x^{2}+y^{2}\right)^{2}=1734\left(x^{2}-y^{2}\right)$ at the point (4,1).

A) $y=1.11 x+5.43$
B) $y=-1.11 x+17$
C) $y=-1.11 x+3.43$
D) $y=-1.11 x+5.43$
E) $\text { None of these }$

Question 38

The mass of the part of a metal rod that lies between its left end and a point x meters to the right is $S=4 x^{2}$ . Find the linear density when x is 3 m.

A) 4
B) 20
C) 24
D) 12
E) 18

Question 39

Find the accumulated amount after 7 years on an investment of $2,000 earning an interest rate of 5% per year compounded continuously. Round to the nearest cent.

A) $2,814.20
B) $2,838.14
C) $14,700.00
D) $14,717.80

Question 40

A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s how fast is the boat approaching the dock when it is m from the dock? Round the result to the nearest hundredth if necessary.

Question 41

A spherical balloon is being inflated. Find the rate of increase of the surface area with respect to the radius r when r = ft.

Question 42

The height (in meters) of a projectile shot vertically upward from a point $5.5$ m above ground level with an initial velocity of 25.48 m/s is $h=5.5+25.48 t-4.9 t^{2}$ after t seconds.

a. When does the projectile reach its maximum height?
b. What is the maximum height?

A) $2.4 \mathrm{~s}$
$34.428 \mathrm{~m}$
B) $2~s$
$32.86 \mathrm{~m}$
C) $2.6 \mathrm{~s}$
$38.624 \mathrm{~m}$
D) $2.8 \mathrm{~s}$
$34.428 \mathrm{~m}$
E) $2.3 \mathrm{~s}$
$34.183 \mathrm{~m}$

Question 43

Suppose that f and g are functions that are differentiable at x = 2 and that f (2) = -1, (2) = 3, g(2) = 3, and (2) = -4. Find .

Question 44

The mass of part of a wire is $x(1+\sqrt{x})$ kilograms, where x is measured in meters from one end of the wire. Find the linear density of the wire when x = 16 m .

A) $6 \mathrm{~kg} / \mathrm{m}$ .
B) $6 \mathrm{~kg} / \mathrm{m}$
C) $1.5 \mathrm{~kg} / \mathrm{m}$
D) $4 \mathrm{~kg} / \mathrm{m}$
E) $\text { None of these }$

Question 45

Differentiate the function. $h(t)=\frac{\ln 6 t}{\ln 12 t}$

A) $\frac{\ln 2}{(\ln 12 t)^{2}}$
B) $\frac{\ln 2}{t(\ln 12 t)^{2}}$
C) $\frac{1}{6 t}-\frac{1}{12 t}$
D) $\frac{\ln 2}{t}$

Question 46

s(t) is the position of a body moving along a coordinate line; s(t) is measured in feet and t in seconds, where . Find the position, velocity, and speed of the body at the indicated time. ; t = 3

Question 47

Newton's Law of Gravitation says that the magnitude F of the force exerted by a body of mass m on a body of mass M is .
Find .

Question 48

Differentiate the function. $g(t)=t^{5} \ln 9 t$

A) $1+\frac{\ln 9 t}{9 t}$
B) $\frac{5}{9} t^{3}$
C) $t^{4}(1+5 \ln 9 t)$
D) $t^{4}\left(\frac{1}{9}+5 \ln 9 t\right)$

Question 49

A company makes computer chips from square wafers of silicon. It wants to keep the side length of a wafer very close to mm. The area is A(x). Find ( ).

Question 50

s(t) is the position of a body moving along a coordinate line; s(t) is measured in feet and t in seconds, where . Find the position, velocity, and speed of the body at the indicated time. ; t = 1

Question 51

Refer to the law of laminar flow. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference $3,500 \text { dynes } / \mathrm{cm}^{2}$ and viscosity $\eta$ =.028.
Find the velocity of the blood at radius r = $0.001$

Question 52

Suppose that f and g are functions that are differentiable at x = -3 and that f (-3) = 3, (-3) = -5, g (-3) = 3, and (-3) = 3. Find .

Question 53

If $f(x)=\frac{x}{\ln x}$ , find $f^{\prime}\left(e^{4}\right)$ .

A) $\frac{3}{16}$
B) $\frac{e^{4}}{9}$
C) $\frac{16}{3}$
D) $\frac{4}{3}$
E) $0.4$

Question 54

Calculate y'. $y=4 \ln \left(x^{2} e^{x}\right)$

A) $y^{\prime}=4\left(\frac{2}{x}+1\right)$
B) $y^{\prime}=x \ln 2 x+x^{2}$
C) $y^{\prime}=4\left(\frac{2}{x}+2\right)$
D) $y^{\prime}=\frac{2 e^{x}+x \ln x^{2}}{x}$
E) $y^{\prime}=3 x^{2}$

Question 55

Find the differential of the function at the indicated number. ;

Question 56

If a tank holds 5000 gallons of water, and that water can drain from the tank in 40 minutes, then Torricelli's Law gives the volume V of water remaining in the tank after t minutes as .
Find the rate at which water is draining from the tank after minutes.

Question 57

In calm waters, the oil spilling from the ruptured hull of a grounded tanker spreads in all directions. Assuming that the polluted area is circular, determine how fast the area is increasing when the radius of the circle is 20 ft and is increasing at the rate of ft/sec. Round to the nearest tenth if necessary.

Question 58

Two chemicals react to form another chemical. Suppose that the amount of chemical formed in time t (in hours) is given by where is measured in pounds.
a. Find the rate at which the chemical is formed when Round to two decimal places.
b. How many pounds of the chemical are formed eventually?

Question 59

s(t) is the position of a body moving along a coordinate line, where , and s(t) is measured in feet and t in seconds.
a. Determine the time(s) and the position(s) when the body is stationary.
b. When is the body moving in the positive direction? In the negative direction?
c. Sketch a schematic showing the position of the body at any time t.

Question 60

The position function of a particle is given by When does the particle reach a velocity of m/s?

Question 61

Find dy/dx by implicit differentiation. $5 x^{2}+9 y^{2}=5$

A) $\frac{5-10 x}{18 y}$
B) $-\frac{5 x}{9 y}$
C) $-\frac{5 x}{9}$
D) $\frac{5-10 x}{18}$

Question 62

Use logarithmic differentiation to find the derivative of the function. $y=x^{6 x}$

A) $y^{\prime}=6(\ln x+1)$
B) $y^{\prime}=6 x^{6 x}(\ln x+1)$
C) $y^{\prime}=-6 x^{6 x}(\ln x+6)$
D) $y^{\prime}=6 x^{6 x}(6 \ln x+1)$
E) $y^{\prime}=x^{x}(\ln 6 x+1)$

Question 63

If $f(t)=\sqrt{9 t+1}$ , find $f^{\prime \prime}(4)$ .

A) $-0.010$
B) $0.015$
C) $-0.033$
D) $-0.22$
E) $0.044$

Question 64

Find $\frac{d^{5}}{d x^{5}}\left(x^{4} \ln x\right)$ .

A) $\frac{d^{5}}{d x^{5}}\left(x^{4} \ln x\right)=\frac{3}{x^{4}}$
B) $\frac{d^{5}}{d x^{5}}\left(x^{4} \ln x\right)=\frac{6}{x^{2}}$
C) $\frac{d^{5}}{d x^{5}}\left(x^{4} \ln x\right)=\frac{1}{x}$
D) $\frac{d^{5}}{d x^{5}}\left(x^{4} \ln x\right)=\frac{6}{x^{4}}$
E) $\frac{d^{5}}{d x^{5}}\left(x^{4} \ln x\right)=\frac{24}{x}$

Question 65

Differentiate the function.

Question 66

Use implicit differentiation to find an equation of the tangent line to the curve at the indicated point. y = sin xy⁶; $\left(\frac{\pi}{2}, 1\right)$

A) x = $\frac{\pi}{2}$
B) y = 6x + 1
C) y = x
D) y = 1

Question 67

Find $d^{2} y / d x^{2}$ in terms of x and y. $x^{6}-y^{6}=-1$

A) $30 x^{4}-30 y^{4}$
B) $\frac{x^{5}}{y^{5}}$
C) $6 x^{5}-6 y^{5}$
D) $\frac{5 x^{4}}{y^{5}}-\frac{5 x^{10}}{y^{11}}$

Question 68

Find $d^{2} y / d x^{2}$ in terms of x and y. $x^{6}-y^{6}=1$

A) $6 x^{5}-6 y^{5}$
B) $\frac{5 x^{4}}{y^{5}}-\frac{5 x^{10}}{y^{11}}$
C) $30 x^{4}-30 y^{4}$
D) $\frac{x^{5}}{y^{5}}$

Question 69

Find dy/dx by implicit differentiation. $e^{x y}-x^{8}+y^{8}=8$

A) $\frac{8 x^{7}-x e^{x y}}{y e^{x y}+8 y^{7}}$
B) $\frac{8 y^{7}-y e^{x y}}{x e^{x y}+8 x^{7}}$
C) $\frac{8 y^{7}-x e^{x y}}{y e^{x y}+8 x^{7}}$
D) $\frac{8 x^{7}-y e^{x y}}{x e^{x y}+8 y^{7}}$

Question 70

Use logarithmic differentiation to find the derivative of the function.

Question 71

Find the tangent line to the ellipse $\frac{x^{2}}{24}+\frac{y^{2}}{6}=1$ at the point $(2,-\sqrt{3})$ .

A) $y=\frac{\sqrt{3}}{3} x-4$
B) $y=\frac{\sqrt{3}}{6} x-\frac{4 \sqrt{3}}{3}$
C) $y=-\sqrt{6} x-3$
D) $y=\sqrt{3} x-4$
E) $\text { None of these }$

Question 72

Use logarithmic differentiation to find the derivative of the function. $y=\sqrt[9]{\frac{x^{2}+1}{x^{2}-1}}$

A) $y^{\prime}=-\frac{9 x}{\left(x^{4}-1\right)} \sqrt[9]{\frac{x^{2}+1}{x^{2}-1}}$
B) $y^{\prime}=\frac{9 x}{4 x^{4}-1} \sqrt[9]{\frac{x^{2}+1}{x^{2}-1}}$
C) $y^{\prime}=-\frac{36 x}{x^{4}-1} \sqrt[9]{\frac{x^{2}+1}{x^{2}-1}}$
D) $y^{\prime}=-\frac{36 x}{x^{4}-1}$
E) $y^{\prime}=-\frac{4 x}{9\left(x^{4}-1\right)} \sqrt[9]{\frac{x^{2}+1}{x^{2}-1}}$

Question 73

Use implicit differentiation to find dy/dx. $\ln x y-y^{3}=9$

A) $\frac{x}{y\left(3 y^{3}-1\right)}$
B) $\frac{y\left(3 y^{3}-1\right)}{x}$
C) $\frac{y}{x\left(3 y^{3}-1\right)}$
D) $\frac{x\left(3 y^{3}-1\right)}{y}$

Question 74

Differentiate the function.

Question 75

Differentiate the function.

Question 76

Find an equation of the tangent line to the curve at .

Question 77

Calculate $y^{\prime }$ . $x y^{5}+x^{4} y=x+3 y$

A) $y^{\prime}=\frac{1-y^{5}-2 x^{4}}{5 x y^{3}+x^{3}-3}$
B) $y^{\prime}=\frac{x y^{4}+3 x-3}{x^{4} y^{3}(5 x-1)}$
C) $y^{\prime}=\frac{-y^{4}-4 x y}{4 x y^{3}+x^{3}}$
D) $y^{\prime}=\frac{1-y^{5}-4 x^{3} y}{5 x y^{4}+x^{4}-3}$
E) none of these

Question 78

Differentiate the function.

Question 79

Use logarithmic differentiation to find the derivative of the function. $y=\sqrt[3]{\frac{x^{8}+1}{x-1}}$

A) $\frac{7 x^{8}-8 x^{7}+1}{3\left(x^{8}+1\right)^{2 / 3}(x-1)^{4 / 3}}$
B) $\frac{7 x^{8}-8 x^{7}-1}{3\left(x^{8}+1\right)^{2 / 3}(x-1)^{4 / 3}}$
C) $\frac{7 x^{8}-8 x^{7}+1}{3\left(x^{8}+1\right)(x-1)}$
D) $\frac{7 x^{8}-8 x^{7}-1}{3\left(x^{8}+1\right)(x-1)}$

Question 80

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. $y \sin 3 x=x \cos 3 y,\left(\frac{\pi}{3}, \frac{\pi}{6}\right)$

A) $y=\frac{x}{3}$
B) $y=\frac{x}{3}+\frac{\pi}{6}$
C) $y=2 x-\frac{3 \pi}{3}$
D) $y=\frac{x}{6}$
E) $y=-\frac{x}{2}+\frac{\pi}{2}$