Question 1

Use logarithmic differentiation to find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ .   $y = \left( x ^ { 3 } + x \right) \sqrt { x ^ { 3 } + 6 }$ ?

Accepted Answer

A)  $\left( \frac { 3 x ^ { 2 } + 1 } { x ^ { 3 } + x } + \frac { 3 x ^ { 2 } } { 2 x ^ { 3 } + 12 } \right)$ 
B)   $\left( x ^ { 3 } + x \right) \sqrt { x ^ { 3 } + 6 } \left( \frac { 1 } { x ^ { 3 } + x } + \frac { 1 } { 2 \left( x ^ { 3 } + 6 \right) } \right)$ 
C)    $\left( x ^ { 3 } + x \right) \sqrt { x ^ { 3 } + 6 } \left( \frac { 3 x ^ { 2 } + 1 } { x ^ { 3 } + x } + \frac { 3 x ^ { 2 } } { 2 \left( x ^ { 3 } + 6 \right) } \right)$ 
D)    $\left( 3 x ^ { 2 } + 1 \right) \left( \frac { 3 x ^ { 2 } + 1 } { x } + \frac { 3 x ^ { 2 } } { 2 x ^ { 3 } + 12 } \right)$ 
E)    $3 x ^ { 2 } \sqrt { x ^ { 3 } + 12 } \left( \frac { 3 x ^ { 2 } + 1 } { x ^ { 3 } + x } + \frac { x ^ { 2 } } { 2 x ^ { 3 } + 6 } \right)$ 
A)  $\left( \frac { 3 x ^ { 2 } + 1 } { x ^ { 3 } + x } + \frac { 3 x ^ { 2 } } { 2 x ^ { 3 } + 12 } \right)$ 
B)   $\left( x ^ { 3 } + x \right) \sqrt { x ^ { 3 } + 6 } \left( \frac { 1 } { x ^ { 3 } + x } + \frac { 1 } { 2 \left( x ^ { 3 } + 6 \right) } \right)$ 
C)    $\left( x ^ { 3 } + x \right) \sqrt { x ^ { 3 } + 6 } \left( \frac { 3 x ^ { 2 } + 1 } { x ^ { 3 } + x } + \frac { 3 x ^ { 2 } } { 2 \left( x ^ { 3 } + 6 \right) } \right)$ 
D)    $\left( 3 x ^ { 2 } + 1 \right) \left( \frac { 3 x ^ { 2 } + 1 } { x } + \frac { 3 x ^ { 2 } } { 2 x ^ { 3 } + 12 } \right)$ 
E)    $3 x ^ { 2 } \sqrt { x ^ { 3 } + 12 } \left( \frac { 3 x ^ { 2 } + 1 } { x ^ { 3 } + x } + \frac { x ^ { 2 } } { 2 x ^ { 3 } + 6 } \right)$

Question 2

Find the equation of the straight line, tangent to $y = e ^ { 7 x } \log _ { 5 } x$ at the point $( 1,0 )$ .

Accepted Answer

A)   $y ( x ) = \frac { e ^ { 7 } } { \ln 5 } x + \frac { e ^ { 7 } } { \ln 5 }$ 
B)    $y ( x ) = \frac { e ^ { 5 } } { \ln 7 } x - \frac { e ^ { 5 } } { \ln 7 }$ 
C)    $y ( x ) = \frac { e ^ { 5 } } { \ln 7 } x + \frac { e ^ { 5 } } { \ln 7 }$ 
D)    $y ( x ) = \frac { e ^ { 7 } } { \ln 5 } x - \frac { e ^ { 7 } } { \ln 5 }$ 
E)  none of these 
A)   $y ( x ) = \frac { e ^ { 7 } } { \ln 5 } x + \frac { e ^ { 7 } } { \ln 5 }$ 
B)    $y ( x ) = \frac { e ^ { 5 } } { \ln 7 } x - \frac { e ^ { 5 } } { \ln 7 }$ 
C)    $y ( x ) = \frac { e ^ { 5 } } { \ln 7 } x + \frac { e ^ { 5 } } { \ln 7 }$ 
D)    $y ( x ) = \frac { e ^ { 7 } } { \ln 5 } x - \frac { e ^ { 7 } } { \ln 5 }$ 
E)  none of these

Question 3

Evaluate the following derivative.
​ $\frac { \mathrm { d } } { \mathrm { d } t } \left( a t ^ { 2 } + q t + r \right)$

Accepted Answer

The answer of Evaluate the following derivative.
​ \(\frac { \mathrm...

Question 4

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ using implicit differentiation.   $\frac { x y } { 2 } - y ^ { 2 } = 9$

Accepted Answer

A)  $\frac { y } { 4 y - x }$ 
B)  $\frac { 1 } { \sqrt { 4 x y } }$ 
C)   $4 y - 2 x$ 
D)   $\frac { 1 } { 2 y - x }$ 
E)   $\frac { y } { 4 x - y }$ 
A)  $\frac { y } { 4 y - x }$ 
B)  $\frac { 1 } { \sqrt { 4 x y } }$ 
C)   $4 y - 2 x$ 
D)   $\frac { 1 } { 2 y - x }$ 
E)   $\frac { y } { 4 x - y }$

Question 5

The Pentagon is planning to build a new satellite that will be spherical. As is typical in these cases, the specifications keep changing, so that the size of the satellite keeps growing. In fact, the radius of the planned satellite is growing 0.9 foot/week. Its cost will be $1,400 per cubic foot. At the point when the plans call for a satellite 8 feet in radius, how fast is the cost growing  (The volume of a solid sphere of radius r is $V = \frac { 4 } { 3 } \pi r ^ { 3 }$ .)

Accepted Answer

A)  $\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { \$ 3,584,000 \pi } { \text { week } }$ 
B)  $\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { \$ 40,320 \pi } { \text { week } }$ 
C)  $\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { \$ 230 \pi } { \text { week } }$ 
D)  $\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { \$ 161,280 \pi } { \text { week } }$ 
E)  $\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { \$ 322,560 \pi } { \text { week } }$ 
A)  $\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { \$ 3,584,000 \pi } { \text { week } }$ 
B)  $\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { \$ 40,320 \pi } { \text { week } }$ 
C)  $\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { \$ 230 \pi } { \text { week } }$ 
D)  $\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { \$ 161,280 \pi } { \text { week } }$ 
E)  $\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { \$ 322,560 \pi } { \text { week } }$

Question 6

Find the derivative of the function.  $s ( x ) = 19 \sqrt { x } + \frac { 33 } { \sqrt { x } }$

Accepted Answer

A)  $s ^ { \prime } ( x ) = \frac { 9.5 } { \sqrt { x } } + \frac { 33 } { x \sqrt { x } }$ 
B)  $s ^ { \prime } ( x ) = \frac { 9.5 } { \sqrt { x } } - \frac { 16.5 } { x \sqrt { x } }$ 
C)  $s ^ { \prime } ( x ) = \frac { 19 } { \sqrt { x } } - \frac { 33 } { x \sqrt { x } }$ 
D)  $s ^ { \prime } ( x ) = \frac { 9.5 } { \sqrt { x } } + \frac { 16.5 } { x \sqrt { x } }$ 
E)  $s ^ { \prime } ( x ) = \frac { 8.5 } { \sqrt { x } } + \frac { 17.5 } { x \sqrt { x } }$ 
A)  $s ^ { \prime } ( x ) = \frac { 9.5 } { \sqrt { x } } + \frac { 33 } { x \sqrt { x } }$ 
B)  $s ^ { \prime } ( x ) = \frac { 9.5 } { \sqrt { x } } - \frac { 16.5 } { x \sqrt { x } }$ 
C)  $s ^ { \prime } ( x ) = \frac { 19 } { \sqrt { x } } - \frac { 33 } { x \sqrt { x } }$ 
D)  $s ^ { \prime } ( x ) = \frac { 9.5 } { \sqrt { x } } + \frac { 16.5 } { x \sqrt { x } }$ 
E)  $s ^ { \prime } ( x ) = \frac { 8.5 } { \sqrt { x } } + \frac { 17.5 } { x \sqrt { x } }$

Question 7

The cost of controlling emissions at a firm goes up rapidly as the amount of emissions reduced goes up. Here is a possible model:   $C ( q ) = 2,500 + 200 q ^ { 2 }$ 
Where q is the reduction in emissions (in pounds of pollutant per day) and C is the daily cost (in dollars) of this reduction. Government clean-air subsidies to the firm are based on the formula
  $S ( q ) = 800 q$ 
Where q is again the reduction in emissions (in pounds per day) and S is the subsidy (in dollars). Calculate the net cost function
 $N ( q ) = C ( q ) - S ( q )$  
Given the cost and subsidy above, and find the value of q that gives the lowest net cost. What is this lowest net cost

Accepted Answer

A)  $q = \$ 2 , N ( q ) = \$ 3,400$ 
B)  $q = \$ 2 , N ( q ) = \$ 1,700$ 
C)  $q = \$ 2.5 , N ( q ) = \$ 1,700$ 
D)  $q = \$ 4 , N ( q ) = \$ 1,700$ 
E)  $q = \$ 2.5 , N ( q ) = \$ 1,750$ 
A)  $q = \$ 2 , N ( q ) = \$ 3,400$ 
B)  $q = \$ 2 , N ( q ) = \$ 1,700$ 
C)  $q = \$ 2.5 , N ( q ) = \$ 1,700$ 
D)  $q = \$ 4 , N ( q ) = \$ 1,700$ 
E)  $q = \$ 2.5 , N ( q ) = \$ 1,750$

Question 8

Compute the indicated derivative using the chain rule.  $y = 7 x + 10 ; \frac { \mathrm { d } x } { \mathrm {~d} y }$

Accepted Answer

A)  $\frac { 1 } { 10 }$ 
B)   $- \frac { 7 } { 10 }$ 
C)  $\frac { 1 } { 7 }$ 
D)  7 
E)  -10 
A)  $\frac { 1 } { 10 }$ 
B)   $- \frac { 7 } { 10 }$ 
C)  $\frac { 1 } { 7 }$ 
D)  7 
E)  -10

Question 9

The cost of controlling emissions at a firm goes up rapidly as the amount of emissions reduced goes up. Here is a possible model: 
 $C ( q ) = 4,500 + 200 q ^ { 2 }$ 
Where q is the reduction in emissions (in pounds of pollutant per day) and C is the daily cost (in dollars) of this reduction. If a firm is currently reducing its emissions by 12 pounds each day, what is the marginal cost of reducing emissions even further

Accepted Answer

A) $3,600 
B)  $4,800 
C)  $9,300 
D)  $1,200 
E)  $2,400 
A) $3,600 
B)  $4,800 
C)  $9,300 
D)  $1,200 
E)  $2,400

Question 10

The monthly sales of Sunny Electronics' new stereo system is given by $S ( x ) = 10 x - x ^ { 2 }$ hundred units per month, x months after its introduction. The price Sunny charges is $p ( x ) = 1,200 - x ^ { 2 }$ dollars per stereo system, x months after its introduction. The revenue Sunny earns then must be $R ( x ) = 100 p ( x ) S ( x )$ . Find the rate of change of revenue 10 months after introduction.
​
Please enter your answer in dollars/month without the units.

Accepted Answer

The answer of The monthly sales of Sunny Electronics' new...

Question 11

The soap bubble I am blowing has a radius that is growing at a rate of 3 cm/s. How fast is the surface area growing when the radius is 10 cm  (The surface area of a sphere of radius r is $S = 4 \pi r ^ { 2 }$ .)

Accepted Answer

A)  $\frac { \mathrm { d } S } { \mathrm {~d} t } = 85 \pi \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$ 
B)  $\frac { \mathrm { d } S } { \mathrm {~d} t } = \frac { 240 } { \pi } \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$ 
C)  $\frac { \mathrm { d } S } { \mathrm {~d} t } = 240 \pi \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$ 
D)  $\frac { \mathrm { d } S } { \mathrm {~d} t } = 243 \pi \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$ 
E)  $\frac { \mathrm { d } S } { \mathrm {~d} t } = 24 \pi \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$ 
A)  $\frac { \mathrm { d } S } { \mathrm {~d} t } = 85 \pi \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$ 
B)  $\frac { \mathrm { d } S } { \mathrm {~d} t } = \frac { 240 } { \pi } \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$ 
C)  $\frac { \mathrm { d } S } { \mathrm {~d} t } = 240 \pi \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$ 
D)  $\frac { \mathrm { d } S } { \mathrm {~d} t } = 243 \pi \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$ 
E)  $\frac { \mathrm { d } S } { \mathrm {~d} t } = 24 \pi \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$

Question 12

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ using implicit differentiation.  $\ln \left( 15 + e ^ { x y } \right) = y$

Accepted Answer

A)  $x + y$ 
B)    $\frac { y e ^ { x y } } { 15 + e ^ { x y } }$ 
C)    $\frac { y } { 1 - x }$ 
D)    $\frac { y e ^ { x y } } { 15 + e ^ { x y } ( 1 - x ) }$ 
E)    $\frac { 1 } { 15 + e ^ { x y } ( 1 - x ) }$ 
A)  $x + y$ 
B)    $\frac { y e ^ { x y } } { 15 + e ^ { x y } }$ 
C)    $\frac { y } { 1 - x }$ 
D)    $\frac { y e ^ { x y } } { 15 + e ^ { x y } ( 1 - x ) }$ 
E)    $\frac { 1 } { 15 + e ^ { x y } ( 1 - x ) }$

Question 13

Calculate $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . You need not expand your answer.   $y = \frac { ( x + 2 ) ( x + 1 ) } { 2 x - 4 }$

Accepted Answer

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

Question 14

Compute the derivative. ? $\frac { \mathrm { d } } { \mathrm { d } t } \left[ \left( t ^ { 2 } - t ^ { 0.5 } \right) \left( t ^ { 0.5 } + t ^ { - 0.5 } \right) \right] _ { t } = 1$

Accepted Answer

A) 3 
B)  0 
C)  1 
D)  7 
E)  -1 
A) 3 
B)  0 
C)  1 
D)  7 
E)  -1

Question 15

Find the derivative of the function.
​
​ $h ( x ) = \ln [ ( 8 x + 6 ) ( 2 x + 1 ) ]$

Accepted Answer

The answer of Find the derivative of the function.
​
​ \(h...

Question 16

Find the derivative of the function.  $r ( x ) = \ln \mid 4 x + e ^ { 4 x }$

Accepted Answer

A)   $\frac { 4 + 4 e ^ { 4 x } } { 4 x + 4 e ^ { 4 x } }$ 
B)    $\frac { 4 + 4 e ^ { 4 x } } { 4 x + e ^ { 4 x } }$ 
C)   $\frac { 4 + e ^ { 4 x } } { 4 x + e ^ { 4 x } }$ 
D)   $\frac { 4 + e ^ { 4 x } } { 4 x + 4 e ^ { 4 x } }$ 
E)  none of these 
A)   $\frac { 4 + 4 e ^ { 4 x } } { 4 x + 4 e ^ { 4 x } }$ 
B)    $\frac { 4 + 4 e ^ { 4 x } } { 4 x + e ^ { 4 x } }$ 
C)   $\frac { 4 + e ^ { 4 x } } { 4 x + e ^ { 4 x } }$ 
D)   $\frac { 4 + e ^ { 4 x } } { 4 x + 4 e ^ { 4 x } }$ 
E)  none of these

Question 17

Find the indicated derivative. ? $y = 5 \sqrt { x } + \frac { 9 } { \sqrt { x } }$ , $x = 4$ when $t = 1$ , $\left. \frac { \mathrm { d } x } { \mathrm {~d} t } \right| _ { t = 1 } = 5$ ; $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } =  $  
Please round the answer to the nearest hundredth.

Accepted Answer

A)  $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } = 3.44$ 
B)  $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } = 34.38$ 
C)  $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } = 11.88$ 
D)  $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } = 0.69$ 
E)  $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } = 6.88$ 
A)  $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } = 3.44$ 
B)  $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } = 34.38$ 
C)  $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } = 11.88$ 
D)  $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } = 0.69$ 
E)  $\left. \frac { \mathrm { d } y } { \mathrm {~d} t } \right| _ { t = 1 } = 6.88$

Question 18

Calculate $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . You need not expand your answer.   $y = \left( \frac { x } { 1.2 } + \frac { 1.2 } { x } \right) \left( x ^ { 2 } + 7 \right)$

Accepted Answer

A)  $2 x \left( \frac { 1 } { 1.2 } - \frac { 1.2 } { x ^ { 2 } } \right)$ 
B)  $\left( \frac { 1 } { 1.2 } - \frac { 1.2 } { x ^ { 2 } } \right) \left( x ^ { 2 } + 7 \right) - 2 x \left( \frac { x } { 1.2 } + \frac { 1.2 } { x } \right)$ 
C)  $2 x \left( \frac { 1 } { 1.2 } - \frac { 1.2 } { x ^ { 2 } } \right) + \left( \frac { x } { 1.2 } + \frac { 1.2 } { x } \right) \left( x ^ { 2 } + 7 \right)$ 
D)  $2 x$ 
E)  $\left( \frac { 1 } { 1.2 } - \frac { 1.2 } { x ^ { 2 } } \right) \left( x ^ { 2 } + 7 \right) + 2 x \left( \frac { x } { 1.2 } + \frac { 1.2 } { x } \right)$ 
A)  $2 x \left( \frac { 1 } { 1.2 } - \frac { 1.2 } { x ^ { 2 } } \right)$ 
B)  $\left( \frac { 1 } { 1.2 } - \frac { 1.2 } { x ^ { 2 } } \right) \left( x ^ { 2 } + 7 \right) - 2 x \left( \frac { x } { 1.2 } + \frac { 1.2 } { x } \right)$ 
C)  $2 x \left( \frac { 1 } { 1.2 } - \frac { 1.2 } { x ^ { 2 } } \right) + \left( \frac { x } { 1.2 } + \frac { 1.2 } { x } \right) \left( x ^ { 2 } + 7 \right)$ 
D)  $2 x$ 
E)  $\left( \frac { 1 } { 1.2 } - \frac { 1.2 } { x ^ { 2 } } \right) \left( x ^ { 2 } + 7 \right) + 2 x \left( \frac { x } { 1.2 } + \frac { 1.2 } { x } \right)$

Question 19

Compute the indicated derivative using the chain rule.  $y = 7 x - 6$ ; $\frac { \mathrm { d } x } { \mathrm {~d} y }$

Accepted Answer

A)  $\frac { 7 } { 6 }$ 
B)  6 
C)  7 
D)  $\frac { 1 } { 6 }$ 
E)  $\frac { 1 } { 7 }$ 
A)  $\frac { 7 } { 6 }$ 
B)  6 
C)  7 
D)  $\frac { 1 } { 6 }$ 
E)  $\frac { 1 } { 7 }$

Question 20

Calculate the derivative of the function.   $\left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 4 }$

Accepted Answer

A)  $g ^ { \prime } ( x ) = \left( - 16 x ^ { 2 } + 8 x + 8 \right) ^ { - 5 }$ 
B)  $g ^ { \prime } ( x ) = - 4 ( 8 x + 2 ) \left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 5 }$ 
C)   $g ^ { \prime } ( x ) = - 32 \left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 5 }$ 
D)    $g ^ { \prime } ( x ) = - 4 ( 8 x + 2 ) \left( 4 x ^ { 2 } + 2 x + 2 \right)$ 
E)    $g ^ { \prime } ( x ) = - 4 \left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 5 }$ 
A)  $g ^ { \prime } ( x ) = \left( - 16 x ^ { 2 } + 8 x + 8 \right) ^ { - 5 }$ 
B)  $g ^ { \prime } ( x ) = - 4 ( 8 x + 2 ) \left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 5 }$ 
C)   $g ^ { \prime } ( x ) = - 32 \left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 5 }$ 
D)    $g ^ { \prime } ( x ) = - 4 ( 8 x + 2 ) \left( 4 x ^ { 2 } + 2 x + 2 \right)$ 
E)    $g ^ { \prime } ( x ) = - 4 \left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 5 }$

Use logarithmic differentiation to find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . $y = \left( x ^ { 3 } + x \right) \sqrt { x ^ { 3 } + 6 }$ ?

Find the equation of the straight line, tangent to $y = e ^ { 7 x } \log _ { 5 } x$ at the point $( 1,0 )$ .

Evaluate the following derivative. $\frac { \mathrm { d } } { \mathrm { d } t } \left( a t ^ { 2 } + q t + r \right)$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ using implicit differentiation. $\frac { x y } { 2 } - y ^ { 2 } = 9$

Find the derivative of the function. $s ( x ) = 19 \sqrt { x } + \frac { 33 } { \sqrt { x } }$

Compute the indicated derivative using the chain rule. $y = 7 x + 10 ; \frac { \mathrm { d } x } { \mathrm {~d} y }$

The soap bubble I am blowing has a radius that is growing at a rate of 3 cm/s. How fast is the surface area growing when the radius is 10 cm (The surface area of a sphere of radius r is $S = 4 \pi r ^ { 2 }$ .)

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ using implicit differentiation. $\ln \left( 15 + e ^ { x y } \right) = y$

Calculate $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . You need not expand your answer. $y = \frac { ( x + 2 ) ( x + 1 ) } { 2 x - 4 }$

Compute the derivative. ? $\frac { \mathrm { d } } { \mathrm { d } t } \left[ \left( t ^ { 2 } - t ^ { 0.5 } \right) \left( t ^ { 0.5 } + t ^ { - 0.5 } \right) \right] _ { t } = 1$

Find the derivative of the function. $h ( x ) = \ln [ ( 8 x + 6 ) ( 2 x + 1 ) ]$

Find the derivative of the function. $r ( x ) = \ln \mid 4 x + e ^ { 4 x }$

Calculate $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . You need not expand your answer. $y = \left( \frac { x } { 1.2 } + \frac { 1.2 } { x } \right) \left( x ^ { 2 } + 7 \right)$

Compute the indicated derivative using the chain rule. $y = 7 x - 6$ ; $\frac { \mathrm { d } x } { \mathrm {~d} y }$

Calculate the derivative of the function. $\left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 4 }$

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Exam 11: Techniques of Differentiation

Use logarithmic differentiation to find dy dx\frac { \mathrm { d } y } { \mathrm {~d} x } dxdy​ . y=(x3+x)x3+6y = \left( x ^ { 3 } + x \right) \sqrt { x ^ { 3 } + 6 }y=(x3+x)x3+6​ ?

Find the equation of the straight line, tangent to y=e7xlog⁡5xy = e ^ { 7 x } \log _ { 5 } xy=e7xlog5​x at the point (1,0)( 1,0 )(1,0) .

Evaluate the following derivative. ​ ddt(at2+qt+r)\frac { \mathrm { d } } { \mathrm { d } t } \left( a t ^ { 2 } + q t + r \right)dtd​(at2+qt+r)

Find dy dx\frac { \mathrm { d } y } { \mathrm {~d} x } dxdy​ using implicit differentiation. xy2−y2=9\frac { x y } { 2 } - y ^ { 2 } = 92xy​−y2=9

Find the derivative of the function. s(x)=19x+33xs ( x ) = 19 \sqrt { x } + \frac { 33 } { \sqrt { x } }s(x)=19x​+x​33​

Compute the indicated derivative using the chain rule. y=7x+10;dx dyy = 7 x + 10 ; \frac { \mathrm { d } x } { \mathrm {~d} y }y=7x+10; dydx​

The soap bubble I am blowing has a radius that is growing at a rate of 3 cm/s. How fast is the surface area growing when the radius is 10 cm (The surface area of a sphere of radius r is S=4πr2S = 4 \pi r ^ { 2 }S=4πr2 .)

Find dy dx\frac { \mathrm { d } y } { \mathrm {~d} x } dxdy​ using implicit differentiation. ln⁡(15+exy)=y\ln \left( 15 + e ^ { x y } \right) = yln(15+exy)=y

Calculate dy dx\frac { \mathrm { d } y } { \mathrm {~d} x } dxdy​ . You need not expand your answer. y=(x+2)(x+1)2x−4y = \frac { ( x + 2 ) ( x + 1 ) } { 2 x - 4 }y=2x−4(x+2)(x+1)​

Compute the derivative. ? ddt[(t2−t0.5)(t0.5+t−0.5)]t=1\frac { \mathrm { d } } { \mathrm { d } t } \left[ \left( t ^ { 2 } - t ^ { 0.5 } \right) \left( t ^ { 0.5 } + t ^ { - 0.5 } \right) \right] _ { t } = 1dtd​[(t2−t0.5)(t0.5+t−0.5)]t​=1

Find the derivative of the function. ​ ​ h(x)=ln⁡[(8x+6)(2x+1)]h ( x ) = \ln [ ( 8 x + 6 ) ( 2 x + 1 ) ]h(x)=ln[(8x+6)(2x+1)]

Find the derivative of the function. r(x)=ln⁡∣4x+e4xr ( x ) = \ln \mid 4 x + e ^ { 4 x }r(x)=ln∣4x+e4x

Calculate dy dx\frac { \mathrm { d } y } { \mathrm {~d} x } dxdy​ . You need not expand your answer. y=(x1.2+1.2x)(x2+7)y = \left( \frac { x } { 1.2 } + \frac { 1.2 } { x } \right) \left( x ^ { 2 } + 7 \right)y=(1.2x​+x1.2​)(x2+7)

Compute the indicated derivative using the chain rule. y=7x−6y = 7 x - 6y=7x−6 ; dx dy\frac { \mathrm { d } x } { \mathrm {~d} y } dydx​

Calculate the derivative of the function. (4x2+2x+2)−4\left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 4 }(4x2+2x+2)−4

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Use logarithmic differentiation to find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . $y = \left( x ^ { 3 } + x \right) \sqrt { x ^ { 3 } + 6 }$ ?

Find the equation of the straight line, tangent to $y = e ^ { 7 x } \log _ { 5 } x$ at the point $( 1,0 )$ .

Evaluate the following derivative. $\frac { \mathrm { d } } { \mathrm { d } t } \left( a t ^ { 2 } + q t + r \right)$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ using implicit differentiation. $\frac { x y } { 2 } - y ^ { 2 } = 9$

Find the derivative of the function. $s ( x ) = 19 \sqrt { x } + \frac { 33 } { \sqrt { x } }$

Compute the indicated derivative using the chain rule. $y = 7 x + 10 ; \frac { \mathrm { d } x } { \mathrm {~d} y }$

The soap bubble I am blowing has a radius that is growing at a rate of 3 cm/s. How fast is the surface area growing when the radius is 10 cm (The surface area of a sphere of radius r is $S = 4 \pi r ^ { 2 }$ .)

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ using implicit differentiation. $\ln \left( 15 + e ^ { x y } \right) = y$

Calculate $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . You need not expand your answer. $y = \frac { ( x + 2 ) ( x + 1 ) } { 2 x - 4 }$

Compute the derivative. ? $\frac { \mathrm { d } } { \mathrm { d } t } \left[ \left( t ^ { 2 } - t ^ { 0.5 } \right) \left( t ^ { 0.5 } + t ^ { - 0.5 } \right) \right] _ { t } = 1$

Find the derivative of the function. $h ( x ) = \ln [ ( 8 x + 6 ) ( 2 x + 1 ) ]$

Find the derivative of the function. $r ( x ) = \ln \mid 4 x + e ^ { 4 x }$

Calculate $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . You need not expand your answer. $y = \left( \frac { x } { 1.2 } + \frac { 1.2 } { x } \right) \left( x ^ { 2 } + 7 \right)$

Compute the indicated derivative using the chain rule. $y = 7 x - 6$ ; $\frac { \mathrm { d } x } { \mathrm {~d} y }$

Calculate the derivative of the function. $\left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 4 }$