ch5

1.Find the derivative of the functionydefined implicitly in terms ofx.

9xy³+4x³y= 1

dy

dx

=

2.Find the derivative of the functionydefined implicitly in terms ofx.

y=8cos(x+y)

dy

dx

=

3.Use implicit differentiation to finddy

dx

.

6x²+3y²=17

dy

dx

=

4.Use implicit differentiation to finddy

dx

.

5x³+3xy²=7x³

dy

dx

=

5.Use implicit differentiation to finddy

dx

.

ysin(xy) =y²+2

dy

dx

=

6.The number of cars produced whenxdollars is spent on labor andydollarsis spent on capital invested by a manufacturer can be modeled by the following equation. (Bothxandyare measured in thousands of dollars.)

30x^1/3y^2/3=360

(a)

Finddy

dx

.

dy

dx

=

Evaluatedy

dx

at the point(27, 8).

(Round your answer to four decimal places.)

2.Find the derivative of the given function.

f(x) =e^x

1x

f'(x) =

Write allx-values (if any) at which the tangent line to the graph would be horizontal. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

x=

2.Find the derivative of the given function.

f(x) =3^x2

f'(x) =

What is the domain off(x)?

(, 0)(0,)

(,)

(0,)

(, 0)

[0,)

What is the domain off'(x)?

(, 0)(0,)

(, 0)

[0,)

(,)

(0,)

Find allx-values at which the function has horizontal tangent lines. (If an answer does not exist, enter DNE.)

x=

3.Find the derivative of the given function.

f(x) =(e^x+e^x)²

9e^x

f'(x) =

What is the domain off(x)?

(,)

[0,)

(0,)

(, 0)

(, 0]

What is the domain off'(x)?

(,)

(, 0]

(0,)

[0,)

(, 0)

Find allx-values at which the function has horizontal tangent lines. (If an answer does not exist, enter DNE.)

x=

4.Find the derivative of the function.

f(x) = ln1 +8

x

f'(x) =

5.Find the derivative of the function.

f(x) = log₉(5+

x

)

f'(x) =

6.Write the first derivative of the given function.

f(x) =[ln(x)]¹⁰

x²

f'(x) =

For whichx-values doesf(x)

have horizontal tangents? (Enter your answers as a comma-separated list.)

x=

7.Findf'(x)

for the function.

f(x) =x²e^x

f'(x) =

8.Findf'(x)

for the function.

f(x) =

e^4x+4x

f'(x) =

9.Findf'(x)

for the function.

f(x) =4^sin(4x)

f'(x) =

10.Findf'(x)

for the function.

f(x) = ln(4x⁹+x)

f'(x) =

11.Findf'(x)

for the function.

f(x) =x²ln(7x)

f'(x) =

12.Find the derivative of the functionydefined implicitly in terms ofx.

ln(cos(y)) =3x+7

y'=

13.Use logarithmic differentiation to finddy

dx

.

y=x+17

3

x²16

dy

dx

=

14.Use the chain rule twice to find the derivative off(x) = tan¹(e^5x).

f'(x)=d

dx

(tan¹(e^5x))

=1

1 +2

d

dx

=1

1 +

=

15.Use logarithmic differentiation to calculate the derivative.

f(x) =1 + 2x²

45x

4

f'(x) =

16.Use logarithmic differentiation to finddy

dx

.

y=x^log₃^(x)

dy

dx

=

4.A balloon is at a height of20meters,and is rising at the constant rate of5 m/sec.A bicyclist passes beneath it, traveling in a straight line at the constant speed of10 m/sec.How fast is the distance between the bicyclist and the balloon increasing2 secondslater?

m/sec

5.Draw and label a diagram to help solve the related-rates problem.

The side of a cube increases at a rate of1

3

m/s.

Find the rate (in m³/s) at which the volume of the cube increases when the side of the cube is7m.

m³/s

6.Draw and label a diagram to help solve the related-rates problem.

The radius of a sphere increases at a rate of8m/s.Find the rate (inm³/s

) at which the volume increases when the radius is14m.

7.A conical tank has height 3 m and radius 2 m at the top. Water flows in at a rate of2.1m³/min.How fast is the water level rising when it is1.1mfrom the bottom of the tank? (Round your answer to three decimal places.)

m/min

8.A trough has ends shaped like isosceles triangles, with width2mand height5m,and the trough is10mlong. Water is being pumped into the trough at a rate of5m³/min.

At what rate (in m/min) does the height of the water change when the water is4mdeep?

9.A tank contains 100 grams of a substance dissolved in a large amount of water. The tank is filtered in such a way that water drains from the tank, leaving the substance behind in the tank. Consider the volume of the dissolved substance to be negligible. At what rate is the concentration (grams/liter) of the substance changing with respect to time in each scenario?

(a) the rate after 5 hours, if the tank contains50L of water initially, and drains at a constant rate of4L/hr?

g/L

hr

(b) the rate at the instant when 20 liters remain, if the water is draining at2.4L/hr at that instant

g/L

hr

(c) the rate in scenario (b), if the unknown substance is also being added at a rate of30g/hr (and there are 100 grams in the tank at that instant)

g/L

hr

10.A boat is pulled in to a dock by a rope with one end attached to the front of the boat and the other end passing through a ring attached to the dock at a point5fthigher than the front of the boat. The rope is being pulled through the ring at the rate of0.4ft/sec.How fast is the boat approaching the dock when13ftof rope is out?

ft/se

3.Finddy

dx

for the function.

y= (tan(x) + sin(x))⁸

dy

dx

=

4.Find the derivative of the given function.

f(x) = (45x³)²

f'(x) =30x2

(5x3+4)3

What is the domain off(x)?

3

4

5

,3

4

5

(,)

x3

4

5

4

5

,4

5

x4

5

What is the domain off'(x)?

x3

4

5

(,)

3

4

5

,3

4

5

x4

5

4

5

,4

5

Find allx-values at which the function has horizontal tangent lines. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

x=

6.Find(f¹)'(a).

f(x) =x³+3x+4,a= 0

(f¹)'(a) =

3.The population, in millions, of arctic flounder in the Atlantic Ocean is modeled by the functionP(t),

wheretis measured in years.

P(t) =9t+8

0.2t²+ 1

(a)

Determine the initial flounder population (in millions).

8

million flounder

(b)

DetermineP'(10)

(in millions of flounder per year). (Round your answer to four decimal places.)

P'(10) =

million flounder/yr

4.Determine the point of discontinuity.

f(x) =x+ 1

6x2

x=

5.Find the value of constantscanddthat make the function below continuous atx=3.

f(x) =x²4xx<3cx=3d+xx>3

c=

d=