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True or False The derivative of a product is the product of the derivatives.

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If \(F( x) =f(x) g(x)\), then \(F' ( x)\) =______.

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True or False \(\dfrac{d}{dx}x^{n}=nx^{n+1}\), for any integer \(n\).

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If \(f\) and \(g\neq 0\) are two differentiable functions, then \(\dfrac{d}{dx}\!\left(\dfrac{f(x)}{g(x)}\right)\) = ______.

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True or False \(f( x) =\dfrac{e^{x}}{x^{2}}\) can be differentiated using the Quotient Rule or by writing \(f(x) =\dfrac{e^{x}}{x^{2}}=x^{-2}e^{x}\) and using the Product Rule.

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If \(g\neq 0\) is a differentiable function, then \(\dfrac{d}{dx} \left[\dfrac{1}{g(x) }\right]\) = ______.

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If \(f(x) =x\), then \(f'' ( x)\) =______.

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When an object in rectilinear motion is modeled by the function \(s=s(t)\), then the acceleration \(a\) of the object at time \(t\) is given by \(a=a(t)\)= ______.

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\(f( x) =xe^{x}\)

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\(f( x) =x^{2}e^{x}\)

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\(f(x)=x^{2}(x^{3}-1)\)

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\(f(x)=x^{4}(x+5)\)

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\(f(x)=(3x^{2}-5)(2x+1)\)

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\(f(x)=(3x-2)(4x+5)\)

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\(s(t)=(2t^{5}-t)(t^{3}-2t+1)\)

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\(F(u)=(u^{4}-3u^{2}+1)(u^{2}-u+2)\)

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\(f( x) =( x^{3}+1) ( e^{x}+1)\)

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\(f( x) =( x^{2}+1) (e^{x}+x)\)

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\(g(s)={\dfrac{2s}{s+1}}\)

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\(F(z)={\dfrac{z+1}{2z}}\)

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\(G(u)={\dfrac{1-2u}{1+2u}}\)

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\(f(w)={\dfrac{1-w^{2}}{1+w^{2}}}\)

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\(f(x)={\dfrac{4x^{2}-2}{3x+4}}\)

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\(f(x)={\dfrac{-3x^{3}-1}{2x^{2}+1}}\)

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\(f( w) =\dfrac{1}{w^{3}-1}\)

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\(g( v) =\dfrac{1}{v^{2}+5v-1}\)

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\(s(t)=t^{-3}\)

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\(G(u)=u^{-4}\)

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\(f(x) =-\dfrac{4}{e^{x}}\)

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\(f( x) =\dfrac{3}{4e^{x}}\)

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\(f(x)={\dfrac{10}{x^{4}}+\dfrac{3}{x^{2}}}\)

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\(f(x)={\dfrac{2}{x^{5}}-\dfrac{3}{x^{3}}}\)

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\(f(x)=3x^{3}-{\dfrac{1}{3x^{2}}}\)

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\(f(x)=x^{5}-{\dfrac{5}{x^{5}}}\)

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\(s(t)={\dfrac{1}{t}-\dfrac{1}{t^{2}}+\dfrac{1}{t^{3}}}\)

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\(s(t)={\dfrac{1}{t+2}+\dfrac{1}{t^{2}}+\dfrac{1}{t^{3}}}\)

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\(f( x) =\dfrac{e^{x}}{x^{2}}\)

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\(f( x) =\dfrac{x^{2}}{e^{x}}\)

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\(f( x) =\dfrac{x^{2}+1}{xe^{x}}\)

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\(f( x) =\dfrac{xe^{x}}{x^{2}-x}\)

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\(f(x)=3x^{2}+x-2\)

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\(f(x)=-5x^{2}-3x\)

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\(f( x)=e^{x}-3\)

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\(f( x) =x-e^{x}\)

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\(f( x) =(x+5)e^{x}\)

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\(f( x) =3x^{4}e^{x}\)

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\(f( x) =(2x+1)(x^{3}+5)\)

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\(f( x) =(3x-5)(x^{2}-2)\)

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\(f(x)=x+\dfrac{1}{x}\)

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\(f(x)=x-\dfrac{1}{x}\)

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\(f(t)=\dfrac{t^{2}-1}{t}\)

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\(f(u)=\dfrac{u+1}{u}\)

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\(f( x) =\dfrac{e^{x}+x}{x}\)

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\(f( x) =\dfrac{e^{x}}{x}\)

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Find \(y^\prime\) and \(y''\) for (a) \(y={\dfrac{1}{x}}\) and (b) \(y={\dfrac{2x-5}{x}}\).

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Find \(\dfrac{dy}{dx}\) and \(\dfrac{d^{2}y}{dx^{2}}\) for (a) \(y={\dfrac{5}{x^{2}}}\) and (b) \(y={\dfrac{2-3x}{x}}\).

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\(s(t) =16t^{2}+20t\)

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\(s( t) =16t^{2}+10t+1\)

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\(s( t) =4.9t^{2}+4t+4\)

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\(s(t) =4.9t^{2}+5t\)

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\(f^{(4)}(x)\) if \(f(x)=x^{3}-3x^{2}+2x-5 \)

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\(f^{(5)}(x)\) if \(f(x)=4x^{3}+x^{2}-1\)

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\({\dfrac{d^{8}}{dt^{8}}\!\left(\dfrac{1}{8}t^{8}-\dfrac{1}{7} t^{7}+t^{5}-t^{3}\right) }\)

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\({\dfrac{d^{6}}{dt^{6}}(t^{6}+5t^{5}-2t+4)}\)

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\(\dfrac{d^{7}}{du^{7}}( e^{u}+u^{2})\)

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\(\dfrac{d^{10}}{du^{10}}( 2e^{u})\)

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\(\dfrac{d^{5}}{dx^{5}}( -e^{x})\)

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\(\dfrac{d^{8}}{dx^{8}}( 12x-e^{x})\)

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\(f(x)={\dfrac{x^{2}}{x-1}}\) at \({\left(\! -1,-\dfrac{1}{2} \right)\! }\)

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\(f(x)={\dfrac{x}{x+1}}\) at \({( 0,0) }\)

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\(f(x)={\dfrac{x^{3}}{x+1}}\) at \({\left( 1,\dfrac{1}{2}\right) }\)

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\(f(x)={\dfrac{x^{2}+1}{x}}\) at \({\left( 2,\dfrac{5}{2}\right)\! }\)

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\(f( x) =( x+1) ( x^{2}-x-11)\)

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\(f( x) =( 3x^{2}-2) ( 2x+1)\)

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\(f(x)={\dfrac{x^{2}}{x+1}}\)

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\(f(x)={\dfrac{x^{2}+1}{x}}\)

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\(f( x) =xe^{x}\)

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\(f( x) =x^{2}e^{x}\)

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\(f( x) =\dfrac{x^{2}-3}{e^{x}}\)

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\(f( x) =\dfrac{e^{x}}{x^{2}+1}\)

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Let \(u( x) =f( x) \cdot g( x)\) and \(v( x) =\dfrac{g( x) }{f( x)}\).

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Let \(F( t) =f( t) \cdot g( t)\) and \(G( t) =\dfrac{g( t) }{f( t) }\).

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Movement of an Object An object is propelled vertically upward from the ground with an initial velocity of \(39.2{\text{m/s}}\). The distance \(s\) (in meters) of the object from the ground after \( t\) seconds is \(s= s(t)= -4.9t^{2}+39.2t\).

1. What is the velocity of the object at time \(t\)?
2. When will the object reach its maximum height?
3. What is the maximum height?
4. What is the acceleration of the object at any time \(t\)?
5. How long is the object in the air?
6. What is the velocity of the object upon impact with the ground?
7. What is the total distance traveled by the object?

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Movement of a Ball A ball is thrown vertically upward from a height of \(6{\text{ft}}\) with an initial velocity of \(80\text{ft/s}\). The distance \(s\) (in feet) of the ball from the ground after \(t\) seconds is \(s= s(t)=6+80t-16t^{2}\).

1. What is the velocity of the ball after 2 seconds?
2. When will the ball reach its maximum height?
3. What is the maximum height the ball reaches?
4. What is the acceleration of the ball at any time \(t\)?
5. How long is the ball in the air?
6. What is the velocity of the ball upon impact with the ground?
7. What is the total distance traveled by the ball?

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Environmental Cost The cost \(C\), in thousands of dollars, for the removal of a pollutant from a certain lake is given by the function \(C( x) =\dfrac{5x}{110-x},\) where \(x\) is the percent of pollutant removed.

1. What is the domain of \(C?\)
2. Graph \(C.\)
3. What is the cost to remove 80% of the pollutant?
4. Find \(C’ ( x) ,\) the rate of change of the cost \(C\) with respect to the amount of pollutant removed.
5. Find the rate of change of the cost for removing 40%, 60%, 80%, and 90% of the pollutant.
6. Interpret the answers found in (e).

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Investing in Fine Art The value \(V\) of a painting \(t\) years after it is purchased is modeled by the function \[ \begin{eqnarray*} V( t) =\dfrac{100t^{2}+50}{t}+400, 1 ≤ t ≤ 5\\[-19pt] \end{eqnarray*} \]

1. Find the rate of change in the value \(V\) with respect to time.
2. What is the rate of change in value after 1 year?
3. What is the rate of change in value after 3 years?
4. Interpret the answers in (b) and (c).

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Drug Concentration The concentration of a drug in a patient’s blood \(t\) hours after injection is given the function \(f( t) =\dfrac{0.4t}{2t^{2}+1}\) (in milligrams per liter).

1. Find the rate of change of the concentration with respect to time.
2. What is the rate of change of the concentration after 10 min? After 30 min? After 1 hour?
3. Interpret the answers found in (b).
4. Graph \(f\) for the first 5 hours after administering the drug.
5. From the graph, approximate the time (in minutes) at which the concentration of the drug is highest. What is the highest concentration of the drug in the patient’s blood?

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Population Growth A population of 1000 bacteria is introduced into a culture and grows in number according to the formula \(P( t) =1000\left( 1+\dfrac{4t}{100+t^{2}}\right)\), where \(t\) is measured in hours.

1. Find the rate of change in population with respect to time.
2. What is the rate of change in population at \(t=1\), \(t=2\), \(t=3\), and \(t=4\)?
3. Interpret the answers found in (b).
4. Graph \(P=P(t)\), \(0 ≤ t ≤ 20\).
5. From the graph, approximate the time (in hours) when the population is the greatest. What is the maximum population of the bacteria in the culture?

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Economics The price-demand function for a popular e-book is given by \(D( p) =\dfrac{100,000}{p^{2}+10p+50},\) \(5 ≤ p ≤ 20,\) where \(D = D(p)\) is the quantity demanded at the price \(p\) dollars.

1. Find \(D’ ( p) \), the rate of change of demand with respect to price.
2. Find \(D’ ( 5)\), \(D’ ( 10)\), and \(D’ ( 15)\).
3. Interpret the results found in (b).

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Intensity of Light The intensity of illumination \(I\) on a surface is inversely proportional to the square of the distance \(r\) from the surface to the source of light. If the intensity is 1000 units when the distance is 1 meter from the light, find the rate of change of the intensity with respect to the distance when the source is 10 meters from the surface.

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Ideal Gas Law The Ideal Gas Law, used in chemistry and thermodynamics, relates the pressure \(p\), the volume \(V\), and the absolute temperature \(T\) (in Kelvin) of a gas, using the equation \(pV=nRT\), where \(n\) is the amount of gas (in moles) and \(R=8.31\) is the ideal gas constant. In an experiment, a spherical gas container of radius \(r\) meters is placed in a pressure chamber and is slowly compressed while keeping its temperature at 273 K.

1. Find the rate of change of the pressure \(p\) with respect to the radius \( r\) of the chamber. (Hint: The volume \(V\) of a sphere is \(V=\dfrac{4}{3 }\pi r^{3}.\))
2. Interpret the sign of the answer found in (a).
3. If the sphere contains 1.0 mol of gas, find the rate of change of the pressure when \(r=\dfrac{1}{4}m\). (Note: The metric unit of pressure is the pascal, \(Pa\)).

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Body Density The density \(\rho\) of an object is its mass \(m\) divided by its volume \(V\); that is, \(\rho =\dfrac{m}{V}\). If a person dives below the surface of the ocean, the water pressure on the diver will steadily increase, compressing the diver and therefore increasing body density. Suppose the diver is modeled as a sphere of radius \(r\).

1. Find the rate of change of the diver’s body density with respect to the radius \(r\) of the sphere. (Hint: The volume \(V\) of a sphere is \(V= \dfrac{4}{3}\pi r^{3}.\))
2. Interpret the sign of the answer found in (a).
3. Find the rate of change of the diver’s body density when the radius is 45\({cm}\) and the mass is \(80{,}000{g}\) (\(80{kg}\)).

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Rectilinear Motion As an object in rectilinear motion moves, its distance \(s\) from the origin at time \(t\) is given by \(s = s(t) =t^{3}-t+1\), where \(s\) is in meters and \(t\) is in seconds.

1. Find the velocity \(v\), acceleration \(a\), jerk \(J,\) and snap \(S\) of the object at time \(t.\)
2. When is the velocity of the object 0 m/s?
3. Find the acceleration of the object at \(t=2\) and at \(t\) = 5.
4. Does the jerk of the object ever equal 0 m/s\(^{3}\)?
5. How would you interpret the snap for this object in rectilinear motion?

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Rectilinear Motion As an object in rectilinear motion moves, its distance \(s\) from the origin at time \(t\) is given by \(s=s(t) =\dfrac{1}{6}t^{4}-t^{2}+\dfrac{1}{2}t\)+4, where \(s\) is in meters and \(t\) is in seconds.

1. Find the velocity \(v\), acceleration \(a\), jerk \(J\), and snap \(S\) of the object at any time \(t.\)
2. Find the velocity of the object at \(t=0\) and at \(t=3\).
3. Find the acceleration of the object at \(t=0\). Interpret your answer.
4. Is the jerk of the object constant? In your own words, explain what the jerk says about the acceleration of the object.
5. How would you interpret the snap for this object in rectilinear motion?

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Elevator Ride Quality The ride quality of an elevator depends on several factors, two of which are acceleration and jerk. In a study of 367 persons riding in a 1600-kg elevator that moves at an average speed of 4 m/s, the majority of riders were comfortable in an elevator with vertical motion given by \[ s(t) =4t+0.8t^{2}+0.333t^{3} \]

1. Find the acceleration that the riders found acceptable.
2. Find the jerk that the riders found acceptable.

Source: Elevator Ride Quality, January 2007, http://www.lift-report.de/index.php/news/176/368/Elevator-Ride-Quality.

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Elevator Ride Quality In a hospital, the effects of high acceleration or jerk may be harmful to patients, so the acceleration and jerk need to be lower than in standard elevators. It has been determined that a 1600-kg elevator that is installed in a hospital and that moves at an average speed of 4 m/s, should have vertical motion \[ s(t) =4t+0.55t^{2}+0.1167t^{3} \]

1. Find the acceleration of a hospital elevator.
2. Find the jerk of a hospital elevator.

Source: Elevator Ride Quality, January 2007, http://www.lift-report.de/index.php/news/176/368/Elevator-Ride-Quality.

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Current Density in a Wire The current density \(J\) in a wire is a measure of how much an electrical current is compressed as it flows through a wire and is modeled by the function \(J(A) =\dfrac{I}{A}\), where \(I\) is the current (in amperes) and \(A\) is the cross-sectional area of the wire. In practice, current density, rather than merely current, is often important. For example, superconductors lose their superconductivity if the current density is too high.

1. As current flows through a wire, it heats the wire, causing it to expand in area \(A\). If a constant current is maintained in a cylindrical wire, find the rate of change of the current density \(J\) with respect to the radius \(r\) of the wire.
2. Interpret the sign of the answer found in (a).
3. Find the rate of change of current density with respect to the radius \(r\) when a current of 2.5 amps flows through a wire of radius \( r=0.50\) millimeter.

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Derivative of a Reciprocal, Function Prove that if a function \(g\) is differentiable, then \(\dfrac{d}{dx}\!\left[\dfrac{1}{g(x)}\right]\) = − \(\dfrac{g^\prime (x)}{[g(x)]^{2}}\), provided \(g(x)\neq 0\).

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Extended Product Rule Show that if \(f,g\), and \(h\) are differentiable functions, then \[ \begin{eqnarray*} \frac{d}{dx}[f(x)g(x)h(x)]&=&f(x)g(x){h}^{\prime }(x)+f(x){g}^{\prime }(x)h(x)\\ &&+{f}^{\prime }(x)g(x)h(x) \end{eqnarray*} \]

From this, deduce that \[ \frac{d}{dx}[f(x)]^{3}=3[f(x)]^{2}f^\prime (x) \]

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\(y=(x^{2}+1)(x-1)(x+5)\)

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\(y=(x-1)(x^{2}+5)(x^{3}-1)\)

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\(y=(x^{4}+1)^{3}\)

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\(y=(x^{3}+1)^{3}\)

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\(y=(3x+1){\left(1+\dfrac{1}{x}\right) (x^{-5}+1)}\)

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\(y={\left(1-\dfrac{1}{x}\right)\!\! \left(1-\dfrac{1}{x^{2}}\right)\!\! \left(1-\dfrac{1}{x^{3}}\right) }\)

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(Further) Extended Product Rule Write a formula for the derivative of the product of four differentiable functions. That is, find a formula for \(\dfrac{d}{dx}[ f_{1}(x) f_{2}(x) f_{3}(x) f_{4}(x)]\). Also find a formula for \(\dfrac{d}{dx}[f(x)]^{4}.\)

185

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If \(f\) and \(g\) are differentiable functions, show that if \(F(x)\)=\(\dfrac{1}{f(x) g(x)}\), then \[ F’ (x)=-F(x)\left[ {\frac{f^\prime (x)}{f(x)}}+{\frac{g^\prime (x)}{g(x)}}\right] \] provided \(f(x) \neq 0,\) \(g(x) \neq 0.\)

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Higher-Order Derivatives If \(f(x)=\dfrac{1}{1-x}\), find a formula for the \(n\)th derivative of \(f\). That is, find \(f^{(n)}(x)\).

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Let \(f(x)=\dfrac{x^{6}-x^{4}+x^{2}}{x^{4}+1}\). Rewrite \(f\) in the form \((x^{4}+1) f(x) = x^{6}-x^{4}+x^{2}\). Now find \(f^\prime (x)\) without using the quotient rule.

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If \(f\) and \(g\) are differentiable functions with \(f\neq -g\), find the derivative of \(\dfrac{fg}{f+g}\).

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\(f(x) =\dfrac{2x}{x+1}\).

1. Use technology to find \(f^\prime (x).\)
2. Simplify \(f^\prime\) to a single fraction using either algebra or a CAS.
3. Use technology to find \(f^{(5)}(x)\). (Hint: Your CAS may have a method for finding higher-order derivatives without finding other derivatives first.)

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Suppose \(f\) and \(g\) have derivatives up to the fourth order. Find the first four derivatives of the product \(fg\) and simplify the answers. In particular, show that the fourth derivative is \[ \dfrac{d^4}{dx^4} (fg) = f^{(4)}g+4f^{(3)}g^{(1)}+6f^{(2)}g^{(2)}+4f^{(1)}g^{(3)}+fg^{(4)} \]

Identify a pattern for the higher-order derivatives of \(fg\).

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Suppose \(f_{1}(x),\ldots ,f_{n}(x)\) are differentiable functions.

1. Find \(\dfrac{d}{dx}[ f_{1}(x)\cdot \cdots \cdot f_{n}(x)].\)
2. Find \(\dfrac{d}{dx}\!\!\left[ {\dfrac{1}{f_{1}(x)\cdot \cdots \cdot f_{n}(x)}}\right]\! .\)

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Let \(a,b,c\), and \(d\) be real numbers. Define \[ \begin{equation*} \left\vert \begin{array}{l@{\quad}l} {a} & {b} \\ {c} & {d} \end{array} \right\vert =ad-bc \end{equation*} \]

This is called a \(2\times 2\) determinant and it arises in the study of linear equations. Let \(f_{1}(x),f_{2}(x),f_{3}(x)\), and \(f_{4}(x)\) be differentiable and let \[ D(x)=\left\vert \begin{array}{l@{\quad}l} {f_{1}(x)} & {f_{2}(x)} \\[3pt] {f_{3}(x)} & {f_{4}(x)} \end{array} \right\vert \]

Show that \[ D^{\prime }(x)=\left\vert \begin{array}{l@{\quad}l} {f_{1}' (x)} & {f_{2}' (x)} \\[3pt] {f_{3}(x)} & {f_{4}(x)} \end{array} \right\vert +\left\vert \begin{array}{l@{\quad}l} {f_{1}(x)} & {f_{2}(x)} \\[3pt] {f_{3}' (x)} & {f_{4}' (x)} \end{array} \right\vert \]

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Let \(f_{0}(x) =x-1\)

\[ f_{1}( x) =1+\dfrac{1}{x-1}\]

\[f_{2}( x) =1+\dfrac{1}{1+\dfrac{1}{x-1}}\]

\[f_{3}( x) =1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{x-1}}} \]

1. Write \(f_{1}\), \(f_{2}\,\) \(f_{3}\), \(f_{4}\) and \(f_{5}\) in the form \(\dfrac{ax+b}{cx+d}\).
2. Using the results from (a), write the sequence of numbers representing the coefficients of \( x\) in the numerator, beginning with \(f_{0}(x) =x-1\).
3. Write the sequence in (b) as a recursive sequence. (Hint: Look at the sums of consecutive terms.)
4. Find \(f_{0}'\), \(f_{1}'\), \(f_{2}'\), \(f_{3}'\), \(f_{4}'\), and \(f_{5}'\).