Question 1

$\text { Find } f ^ { tt} ( x ) \text { if } f ( x ) = ( 7 + 4 x ) e ^ { - 4 x }$

Accepted Answer

A)  $f ^ { tt } ( x ) = - 80 ( 7 + 4 x ) e ^ { - 4 x }$ 
B)  $f ^ { tt } ( x ) = ( - 80 - 64 x ) e ^ { - 4 x }$ 
C)  $f ^ { tt} ( x ) = - ( 24 + 16 x ) e ^ { - 4 x }$ 
D)  $f ^ { tt} ( x ) = ( 80 - 64 x ) e ^ { - 4 x }$ 
E)  $f ^ { tt } ( x ) = ( 80 + 64 x ) e ^ { - 4 x }$ 
A)  $f ^ { tt } ( x ) = - 80 ( 7 + 4 x ) e ^ { - 4 x }$ 
B)  $f ^ { tt } ( x ) = ( - 80 - 64 x ) e ^ { - 4 x }$ 
C)  $f ^ { tt} ( x ) = - ( 24 + 16 x ) e ^ { - 4 x }$ 
D)  $f ^ { tt} ( x ) = ( 80 - 64 x ) e ^ { - 4 x }$ 
E)  $f ^ { tt } ( x ) = ( 80 + 64 x ) e ^ { - 4 x }$

Question 2

airplane flies at an altitude of 6 miles towards a point directly over an observer. Consider $\theta$ and $x$ as shown in the following figure. The speed of the plane is 400 miles per hour.
Find $\frac { d \theta } { d t }$ when $x = 10$. Round your answer to three decimal places.

Accepted Answer

A)  $-17.647 \mathrm { rad } / \mathrm { hr }$ 
B)  $2.941 \mathrm { rad } / \mathrm { hr }$ 
C)  $17.647 \mathrm { rad } / \mathrm { hr }$ 
D)  $0.044 \mathrm { rad } / \mathrm { hr }$ 
E)  $- 8.824 \mathrm { rad } / \mathrm { hr }$ 
A)  $-17.647 \mathrm { rad } / \mathrm { hr }$ 
B)  $2.941 \mathrm { rad } / \mathrm { hr }$ 
C)  $17.647 \mathrm { rad } / \mathrm { hr }$ 
D)  $0.044 \mathrm { rad } / \mathrm { hr }$ 
E)  $- 8.824 \mathrm { rad } / \mathrm { hr }$

Question 3

$$\text { Find the derivative of the function } f ( x ) = \sqrt { 7 x - 3 } \text { using the limiting process. }$$

Accepted Answer

A)  $f ^ { t } ( x ) = \frac { 7 } { 2 \sqrt { 7 x - 3 } }$ 
B)  $f ^ { t } ( x ) = - \frac { 7 } { 2 \sqrt { 7 x - 3 } }$ 
C)  $f ^ { t } ( x ) = - \frac { 7 x } { \sqrt { 7 x - 3 } }$ 
D)  $f ^ {t } ( x ) = \frac { 7 } { 2 } \sqrt { 7 x - 3 }$ 
E)  $f ^ { t } ( x ) = - \frac { 7 } { \sqrt { 7 x - 3 } }$ 
A)  $f ^ { t } ( x ) = \frac { 7 } { 2 \sqrt { 7 x - 3 } }$ 
B)  $f ^ { t } ( x ) = - \frac { 7 } { 2 \sqrt { 7 x - 3 } }$ 
C)  $f ^ { t } ( x ) = - \frac { 7 x } { \sqrt { 7 x - 3 } }$ 
D)  $f ^ {t } ( x ) = \frac { 7 } { 2 } \sqrt { 7 x - 3 }$ 
E)  $f ^ { t } ( x ) = - \frac { 7 } { \sqrt { 7 x - 3 } }$

Question 4

$\text { Differentiate the function } f ( x ) = \ln \left( \frac { e ^ { 5 x } + 1 } { e ^ { 2 x } + 1 } \right) \text {. }$

Accepted Answer

A)  $\frac { 2 e ^ { 2 x } + 1 } { 5 e ^ { 2 x } + 1 }$ 
B)  $\frac { 5 e ^ { 5 x } } { e ^ { 5 x } + 1 } - \frac { 2 e ^ { 2 x } } { e ^ { 2 x } + 1 }$ 
C)  $\frac { 5 e ^ { 5 x } } { e ^ { 5 x } + 1 } + \frac { 2 e ^ { 2 x } } { e ^ { 2 x } + 1 }$ 
D)  $\frac { e ^ { 2 x } + 1 } { e ^ { 5 x } + 1 }$ 
E)  $\frac { e ^ { 5 x } } { e ^ { 5 x } + 1 } + \frac { e ^ { 2 x } } { e ^ { 2 x } + 1 }$ 
A)  $\frac { 2 e ^ { 2 x } + 1 } { 5 e ^ { 2 x } + 1 }$ 
B)  $\frac { 5 e ^ { 5 x } } { e ^ { 5 x } + 1 } - \frac { 2 e ^ { 2 x } } { e ^ { 2 x } + 1 }$ 
C)  $\frac { 5 e ^ { 5 x } } { e ^ { 5 x } + 1 } + \frac { 2 e ^ { 2 x } } { e ^ { 2 x } + 1 }$ 
D)  $\frac { e ^ { 2 x } + 1 } { e ^ { 5 x } + 1 }$ 
E)  $\frac { e ^ { 5 x } } { e ^ { 5 x } + 1 } + \frac { e ^ { 2 x } } { e ^ { 2 x } + 1 }$

Question 5

$\text { Use logarithmic differentiation to find } \frac { d y } { d x } \text {. }$ $y = \frac { 5 x - 3 } { ( 3 x + 2 ) ^ { 3 } }$

Accepted Answer

A)  $\frac { - 30 x + 37 } { ( 3 x + 2 ) ^ { 5 } }$ 
B)  $\frac { - 30 x - 17 } { ( 3 x + 2 ) ^ { 5 } }$ 
C)  $\frac { 60 x + 37 } { ( 3 x + 2 ) ^ { 4 } }$ 
D)  $\frac { - 30 x - 17 } { ( 3 x + 2 ) ^ { 2 } }$ 
E)  $\frac { - 30 x + 37 } { ( 3 x + 2 ) ^ { 4 } }$ 
A)  $\frac { - 30 x + 37 } { ( 3 x + 2 ) ^ { 5 } }$ 
B)  $\frac { - 30 x - 17 } { ( 3 x + 2 ) ^ { 5 } }$ 
C)  $\frac { 60 x + 37 } { ( 3 x + 2 ) ^ { 4 } }$ 
D)  $\frac { - 30 x - 17 } { ( 3 x + 2 ) ^ { 2 } }$ 
E)  $\frac { - 30 x + 37 } { ( 3 x + 2 ) ^ { 4 } }$

Question 6

Suppose that the total number of arrests T (in thousands) for all males ages 14 to 27 in 2006 is approximated by the model $T = 0.602 x ^ { 2 } - 41.44 x ^ { 2 } + 922.8 x - 6330,14 \leq x \leq 27$ where $x$ is the age in years (see figure). Approximate the two ages to one decimal place that had total arrests of 275 thousand.

Accepted Answer

A)  $15.7,23.8$ 
B)  $18.2 ; 21.6$ 
C)  $16.7 ; 22.8$ 
D)  $15.7 ; 22.8$ 
E)  $16.7 ; 21.6$ 
A)  $15.7,23.8$ 
B)  $18.2 ; 21.6$ 
C)  $16.7 ; 22.8$ 
D)  $15.7 ; 22.8$ 
E)  $16.7 ; 21.6$

Question 7

$\text { Find the derivative of the function } f ( x ) = \frac { x ^ { 5 } - 9 } { x ^ { 4 } } \text {. }$

Accepted Answer

A)  $f ^ { t } ( x ) = 1 + \frac { 36 } { x ^ { 5 } }$ 
B)  $f ^ { t } ( x ) = 1 - \frac { 36 } { x ^ { 5 } }$ 
C)  $f ^ { t} ( x ) = 1 + \frac { 4 } { x ^ { 5 } }$ 
D)  $f ^ { t } ( x ) = 1 - \frac { 9 } { x ^ { 5 } }$ 
E)  $f ^ {t } ( x ) = 1 + \frac { 9 } { x ^ { 5 } }$ 
A)  $f ^ { t } ( x ) = 1 + \frac { 36 } { x ^ { 5 } }$ 
B)  $f ^ { t } ( x ) = 1 - \frac { 36 } { x ^ { 5 } }$ 
C)  $f ^ { t} ( x ) = 1 + \frac { 4 } { x ^ { 5 } }$ 
D)  $f ^ { t } ( x ) = 1 - \frac { 9 } { x ^ { 5 } }$ 
E)  $f ^ {t } ( x ) = 1 + \frac { 9 } { x ^ { 5 } }$

Question 8

$\text { Use the Quotient Rule to differentiate the function } f ( x ) = \frac { \sin x } { x ^ { 2 } + 3 } \text {. }$

Accepted Answer

A) 
$$f ^ {t } ( x ) = \frac { \left( 3 + x ^ { 2 } \right) \cos x + 2 x \sin x } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$$ 
B) 
$$f ^ { t } ( x ) = \frac { \left( 3 + x ^ { 2 } \right) \cos x - 2 x \sin x } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$$ 
C) 
$$f ^ { t } ( x ) = \frac { ( 3 + x ) \cos x - 2 x \sin x } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$$ 
D) 
$$f ^ { t } ( x ) = \frac { \left( 3 - x ^ { 2 } \right) \cos x - 2 x \sin x } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$$ 
E) 
$$f ^ { t} ( x ) = \frac { \left( 3 + x ^ { 2 } \right) \cos x - 2 x \sin x } { \left( x ^ { 2 } + 3 \right) }$$ 
A) 
$$f ^ {t } ( x ) = \frac { \left( 3 + x ^ { 2 } \right) \cos x + 2 x \sin x } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$$ 
B) 
$$f ^ { t } ( x ) = \frac { \left( 3 + x ^ { 2 } \right) \cos x - 2 x \sin x } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$$ 
C) 
$$f ^ { t } ( x ) = \frac { ( 3 + x ) \cos x - 2 x \sin x } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$$ 
D) 
$$f ^ { t } ( x ) = \frac { \left( 3 - x ^ { 2 } \right) \cos x - 2 x \sin x } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$$ 
E) 
$$f ^ { t} ( x ) = \frac { \left( 3 + x ^ { 2 } \right) \cos x - 2 x \sin x } { \left( x ^ { 2 } + 3 \right) }$$

Question 9

$\text { Use implicit differentiation to find } \frac { d y } { d x } \text {. }$ $7 x ^ { 2 } + 8 \ln x y = 14$

Accepted Answer

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

Question 10

Find the slope $m \text { of the line tangent to the graph of the function } f ( x ) = 2 - 7 x \text { at the }$ point $( - 1,9 )$

Accepted Answer

A)  $m = - 7$ 
B)  $m = - 2$ 
C)  $m = 2$ 
D)  $m = 7$ 
E)  $m = - 9$ 
A)  $m = - 7$ 
B)  $m = - 2$ 
C)  $m = 2$ 
D)  $m = 7$ 
E)  $m = - 9$

Question 11

$\text { Find the derivative of the function } f ( x ) = 10 \arcsin ( x - 9 ) \text {. }$

Accepted Answer

A) $f ^ { t } ( x ) = \frac { 10 } { \sqrt { ( x - 9 ) ^ { 2 } - 1 } }$ 
B)  $f ^ { t } ( x ) = \frac { 10 } { ( x - 9 ) ^ { 2 } - 1 }$ 
C)  $f ^ { t } ( x ) = \frac { 10 } { ( x - 9 ) ^ { 2 } + 1 }$ 
D)  $f ^ { t } ( x ) = \frac { 10 } { \sqrt { 18 x - x ^ { 2 } - 80 } }$ 
E) $f ^ {t } ( x ) = \frac { 10 } { \sqrt { 1 + ( x - 9 ) ^ { 2 } } }$ 
A) $f ^ { t } ( x ) = \frac { 10 } { \sqrt { ( x - 9 ) ^ { 2 } - 1 } }$ 
B)  $f ^ { t } ( x ) = \frac { 10 } { ( x - 9 ) ^ { 2 } - 1 }$ 
C)  $f ^ { t } ( x ) = \frac { 10 } { ( x - 9 ) ^ { 2 } + 1 }$ 
D)  $f ^ { t } ( x ) = \frac { 10 } { \sqrt { 18 x - x ^ { 2 } - 80 } }$ 
E) $f ^ {t } ( x ) = \frac { 10 } { \sqrt { 1 + ( x - 9 ) ^ { 2 } } }$

Question 12

Find the slope of the graph of the function at the given value. $f ( x ) = 2 x ^ { 2 } + 5 x - \frac { 7 } { x ^ { 2 } }$ when $x = - 4$

Accepted Answer

A)  $f ^ { t } ( - 4 ) = \frac { 359 } { 8 }$ 
B)  $f ^ { t } ( - 4 ) = \frac { 359 } { 2 }$ 
C)  $f ^ { t } ( - 4 ) = - \frac { 359 } { 8 }$ 
D)  $f ^ { t} ( - 4 ) = - \frac { 359 } { 32 }$ 
E)  $f ^ { t } ( - 4 ) = \frac { 359 } { 32 }$ 
A)  $f ^ { t } ( - 4 ) = \frac { 359 } { 8 }$ 
B)  $f ^ { t } ( - 4 ) = \frac { 359 } { 2 }$ 
C)  $f ^ { t } ( - 4 ) = - \frac { 359 } { 8 }$ 
D)  $f ^ { t} ( - 4 ) = - \frac { 359 } { 32 }$ 
E)  $f ^ { t } ( - 4 ) = \frac { 359 } { 32 }$

Question 13

The displacement from equilibrium of an object in harmonic motion on the end of a spring is $y = \frac { 1 } { 6 } \cos 12 t - \frac { 1 } { 2 } \sin 8 t$ where $y$ is measured in feet and $t$ is the time in seconds. Determine the velocity of the object when $t = \pi / 8$. Round your answer to two decimal places.

Accepted Answer

A)  $8.00 \mathrm { ft } / \mathrm { sec }$ 
B)  $4.68 \mathrm { ft } / \mathrm { sec }$ 
C)  $6.00 \mathrm { ft } / \mathrm { sec }$ 
D)  $1.00 \mathrm { ft } / \mathrm { sec }$ 
E)  $5.00 \mathrm { ft } / \mathrm { sec }$ 
A)  $8.00 \mathrm { ft } / \mathrm { sec }$ 
B)  $4.68 \mathrm { ft } / \mathrm { sec }$ 
C)  $6.00 \mathrm { ft } / \mathrm { sec }$ 
D)  $1.00 \mathrm { ft } / \mathrm { sec }$ 
E)  $5.00 \mathrm { ft } / \mathrm { sec }$

Question 14

A population of 620 bacteria is introduced into a culture and grows in number according to the equation $P ( t ) = 620 \left( 1 + \frac { 4 t } { 34 + t ^ { 2 } } \right)$ where $t$ is measured in hours. Find the rate at which the population is growing when $t = 2$. Round your answer to two decimal places.

Accepted Answer

A)  $226.7$ bacteria per hour 
B)  $68.89$ bacteria per hour 
C)  $65.26$ bacteria per hour 
D)  $51.52$ bacteria per hour 
E)  $61.23$ bacteria per hour 
A)  $226.7$ bacteria per hour 
B)  $68.89$ bacteria per hour 
C)  $65.26$ bacteria per hour 
D)  $51.52$ bacteria per hour 
E)  $61.23$ bacteria per hour

Question 15

$\text { Find } \frac { d y } { d x } \text { by implicit differentiation. }$ $x ^ { 2 } + y ^ { 2 } = 25$

Accepted Answer

A)  $\frac { d y } { d x } = \frac { x } { y }$ 
B)  $\frac { d y } { d x } = - \frac { x } { y }$ 
C)  $\frac { d y } { d x } = \frac { y } { x }$ 
D)  $\frac { d y } { d x } = - \frac { y } { x }$ 
E)  $\frac { d y } { d x } = - \frac { x } { y ^ { 2 } }$ 
A)  $\frac { d y } { d x } = \frac { x } { y }$ 
B)  $\frac { d y } { d x } = - \frac { x } { y }$ 
C)  $\frac { d y } { d x } = \frac { y } { x }$ 
D)  $\frac { d y } { d x } = - \frac { y } { x }$ 
E)  $\frac { d y } { d x } = - \frac { x } { y ^ { 2 } }$

Question 16

Newton's Method to approximate the zero(s) of the function $f ( x ) = x ^ { 5 } + 4 x + 1$ accurate to three decimal places.

Accepted Answer

A) $0.250$ 
B)  $- 0.250$ 
C)  $- 0.292$ 
D)  $0.281$ 
E)  $0.197$ 
A) $0.250$ 
B)  $- 0.250$ 
C)  $- 0.292$ 
D)  $0.281$ 
E)  $0.197$

Question 17

$\text { Find the second derivative of the function } f ( x ) = 8 x ^ { \frac { 5 } { 9 } } \text {. }$

Accepted Answer

A) 
$f ^ { tt } ( x ) = \frac { - 160 } { 81 } x ^ { \frac { 4 } { 9 } }$ 
B) 
$f ^ {tt } ( x ) = \frac { 5 } { 81 } x ^ { \frac { - 13 } { 9 } }$ 
C) 
$f ^ {tt } ( x ) = \frac { - 160 } { 81 } x ^ { \frac { - 13 } { 9 } }$ 
D) 
$f ^ { tt } ( x ) = \frac { 160 } { 81 } x ^ { \frac { - 13 } { 9 } }$ 
E) 
$f ^ { tt} ( x ) = 8 x ^ { \frac { - 13 } { 9 } }$ 
A) 
$f ^ { tt } ( x ) = \frac { - 160 } { 81 } x ^ { \frac { 4 } { 9 } }$ 
B) 
$f ^ {tt } ( x ) = \frac { 5 } { 81 } x ^ { \frac { - 13 } { 9 } }$ 
C) 
$f ^ {tt } ( x ) = \frac { - 160 } { 81 } x ^ { \frac { - 13 } { 9 } }$ 
D) 
$f ^ { tt } ( x ) = \frac { 160 } { 81 } x ^ { \frac { - 13 } { 9 } }$ 
E) 
$f ^ { tt} ( x ) = 8 x ^ { \frac { - 13 } { 9 } }$

Question 18

$\text { Find the second derivative of the function } f ( x ) = x ^ { 4 } \sec x \text {. }$

Accepted Answer

A)  $f ^ { tt} ( x ) = x ^ { 2 } \sec x \left( 12 + x \sec ^ { 2 } x + 8 x \tan x + x ^ { 2 } \tan ^ { 2 } x \right)$ 
B)  $f ^ {tt } ( x ) = x ^ { 2 } \sec x \left( 12 + x ^ { 2 } \sec ^ { 2 } x + 8 x \tan x + x ^ { 2 } \tan ^ { 2 } x \right)$ 
C)  $f ^ { tt } ( x ) = x ^ { 2 } \sec x \left( 12 + x ^ { 2 } \sec ^ { 2 } x + 4 x \tan x + x ^ { 2 } \tan ^ { 2 } x \right)$ 
D)  $f ^ { tt } ( x ) = x ^ { 2 } \sec x \left( 12 + x ^ { 2 } \sec ^ { 2 } x + 8 x \tan x + x ^ { 2 } \tan x \right)$ 
E)  $f ^ {tt} ( x ) = x ^ { 2 } \sec x \left( 12 + x ^ { 2 } \sec ^ { 2 } x + 8 x \tan x + x \tan ^ { 2 } x \right)$ 
A)  $f ^ { tt} ( x ) = x ^ { 2 } \sec x \left( 12 + x \sec ^ { 2 } x + 8 x \tan x + x ^ { 2 } \tan ^ { 2 } x \right)$ 
B)  $f ^ {tt } ( x ) = x ^ { 2 } \sec x \left( 12 + x ^ { 2 } \sec ^ { 2 } x + 8 x \tan x + x ^ { 2 } \tan ^ { 2 } x \right)$ 
C)  $f ^ { tt } ( x ) = x ^ { 2 } \sec x \left( 12 + x ^ { 2 } \sec ^ { 2 } x + 4 x \tan x + x ^ { 2 } \tan ^ { 2 } x \right)$ 
D)  $f ^ { tt } ( x ) = x ^ { 2 } \sec x \left( 12 + x ^ { 2 } \sec ^ { 2 } x + 8 x \tan x + x ^ { 2 } \tan x \right)$ 
E)  $f ^ {tt} ( x ) = x ^ { 2 } \sec x \left( 12 + x ^ { 2 } \sec ^ { 2 } x + 8 x \tan x + x \tan ^ { 2 } x \right)$

Question 19

$\text { Find the derivative of the function } \frac { \sin x } { 2 x + \cos x } \text {. Simplify your answer. }$

Accepted Answer

A) 
$\frac { 2 x \cos x - \cos ^ { 2 } x - 2 \sin x + \sin ^ { 2 } x } { ( 2 x + \cos x ) ^ { 2 } }$ 
B) 
$\frac { \cos x } { ( 2 - \sin x ) ^ { 2 } }$ 
C)  $\frac { \cos x } { 2 - \sin x }$ 
D)  $\frac { 1 + 2 x \cos x - 2 \sin x } { ( 2 x + \cos x ) ^ { 2 } }$ 
E)  $\frac { 2 x \cos x + \cos ^ { 2 } x - 2 \sin x - \sin ^ { 2 } x } { ( 2 x + \cos x ) ^ { 2 } }$ 
A) 
$\frac { 2 x \cos x - \cos ^ { 2 } x - 2 \sin x + \sin ^ { 2 } x } { ( 2 x + \cos x ) ^ { 2 } }$ 
B) 
$\frac { \cos x } { ( 2 - \sin x ) ^ { 2 } }$ 
C)  $\frac { \cos x } { 2 - \sin x }$ 
D)  $\frac { 1 + 2 x \cos x - 2 \sin x } { ( 2 x + \cos x ) ^ { 2 } }$ 
E)  $\frac { 2 x \cos x + \cos ^ { 2 } x - 2 \sin x - \sin ^ { 2 } x } { ( 2 x + \cos x ) ^ { 2 } }$

Question 20

$\text { Find } \frac { d y } { d x } \text { by implicit differentiation. }$ $x ^ { 2 } + 5 x + 9 x y - y ^ { 2 } = 4$

Accepted Answer

A)  $\frac { d y } { d x } = \frac { 2 x - 5 + 9 y } { 2 y - 9 x }$ 
B)  $\frac { d y } { d x } = \frac { 2 x + 5 + 9 y } { 2 x - 9 y }$ 
C)  $\frac { d y } { d x } = \frac { x + 5 + 9 y } { y - 9 x }$ 
D)  $\frac { d y } { d x } = \frac { 2 x + 5 + 9 y } { 2 y - 9 x }$ 
E)  $\frac { d y } { d x } = \frac { 2 x + 5 - 9 y } { 2 y - 9 x }$ 
A)  $\frac { d y } { d x } = \frac { 2 x - 5 + 9 y } { 2 y - 9 x }$ 
B)  $\frac { d y } { d x } = \frac { 2 x + 5 + 9 y } { 2 x - 9 y }$ 
C)  $\frac { d y } { d x } = \frac { x + 5 + 9 y } { y - 9 x }$ 
D)  $\frac { d y } { d x } = \frac { 2 x + 5 + 9 y } { 2 y - 9 x }$ 
E)  $\frac { d y } { d x } = \frac { 2 x + 5 - 9 y } { 2 y - 9 x }$

$\text { Find } f ^ { tt} ( x ) \text { if } f ( x ) = ( 7 + 4 x ) e ^ { - 4 x }$

$\text { Find the derivative of the function } f ( x ) = \sqrt { 7 x - 3 } \text { using the limiting process. }$

$\text { Differentiate the function } f ( x ) = \ln \left( \frac { e ^ { 5 x } + 1 } { e ^ { 2 x } + 1 } \right) \text {. }$

$\text { Use logarithmic differentiation to find } \frac { d y } { d x } \text {. }$ $y = \frac { 5 x - 3 } { ( 3 x + 2 ) ^ { 3 } }$

$\text { Find the derivative of the function } f ( x ) = \frac { x ^ { 5 } - 9 } { x ^ { 4 } } \text {. }$

$\text { Use the Quotient Rule to differentiate the function } f ( x ) = \frac { \sin x } { x ^ { 2 } + 3 } \text {. }$

$\text { Use implicit differentiation to find } \frac { d y } { d x } \text {. }$ $7 x ^ { 2 } + 8 \ln x y = 14$

Find the slope $m \text { of the line tangent to the graph of the function } f ( x ) = 2 - 7 x \text { at the }$ point $( - 1,9 )$

$\text { Find the derivative of the function } f ( x ) = 10 \arcsin ( x - 9 ) \text {. }$

Find the slope of the graph of the function at the given value. $f ( x ) = 2 x ^ { 2 } + 5 x - \frac { 7 } { x ^ { 2 } }$ when $x = - 4$

$\text { Find } \frac { d y } { d x } \text { by implicit differentiation. }$ $x ^ { 2 } + y ^ { 2 } = 25$

Newton's Method to approximate the zero(s) of the function $f ( x ) = x ^ { 5 } + 4 x + 1$ accurate to three decimal places.

$\text { Find the second derivative of the function } f ( x ) = 8 x ^ { \frac { 5 } { 9 } } \text {. }$

$\text { Find the second derivative of the function } f ( x ) = x ^ { 4 } \sec x \text {. }$

$\text { Find the derivative of the function } \frac { \sin x } { 2 x + \cos x } \text {. Simplify your answer. }$

$\text { Find } \frac { d y } { d x } \text { by implicit differentiation. }$ $x ^ { 2 } + 5 x + 9 x y - y ^ { 2 } = 4$

Graphs and Models

A Preview of Calculus

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 3: The Derivative and the Tangent Line Problem

Find ftt(x) if f(x)=(7+4x)e−4x\text { Find } f ^ { tt} ( x ) \text { if } f ( x ) = ( 7 + 4 x ) e ^ { - 4 x } Find ftt(x) if f(x)=(7+4x)e−4x

Find the derivative of the function f(x)=7x−3 using the limiting process. \text { Find the derivative of the function } f ( x ) = \sqrt { 7 x - 3 } \text { using the limiting process. } Find the derivative of the function f(x)=7x−3​ using the limiting process.

Differentiate the function f(x)=ln⁡(e5x+1e2x+1). \text { Differentiate the function } f ( x ) = \ln \left( \frac { e ^ { 5 x } + 1 } { e ^ { 2 x } + 1 } \right) \text {. } Differentiate the function f(x)=ln(e2x+1e5x+1​).

Use logarithmic differentiation to find dydx. \text { Use logarithmic differentiation to find } \frac { d y } { d x } \text {. } Use logarithmic differentiation to find dxdy​. y=5x−3(3x+2)3y = \frac { 5 x - 3 } { ( 3 x + 2 ) ^ { 3 } }y=(3x+2)35x−3​

Find the derivative of the function f(x)=x5−9x4. \text { Find the derivative of the function } f ( x ) = \frac { x ^ { 5 } - 9 } { x ^ { 4 } } \text {. } Find the derivative of the function f(x)=x4x5−9​.

Use the Quotient Rule to differentiate the function f(x)=sin⁡xx2+3. \text { Use the Quotient Rule to differentiate the function } f ( x ) = \frac { \sin x } { x ^ { 2 } + 3 } \text {. } Use the Quotient Rule to differentiate the function f(x)=x2+3sinx​.

Use implicit differentiation to find dydx. \text { Use implicit differentiation to find } \frac { d y } { d x } \text {. } Use implicit differentiation to find dxdy​. 7x2+8ln⁡xy=147 x ^ { 2 } + 8 \ln x y = 147x2+8lnxy=14

Find the slope m of the line tangent to the graph of the function f(x)=2−7x at the m \text { of the line tangent to the graph of the function } f ( x ) = 2 - 7 x \text { at the }m of the line tangent to the graph of the function f(x)=2−7x at the point (−1,9)( - 1,9 )(−1,9)

Find the derivative of the function f(x)=10arcsin⁡(x−9). \text { Find the derivative of the function } f ( x ) = 10 \arcsin ( x - 9 ) \text {. } Find the derivative of the function f(x)=10arcsin(x−9).

Find the slope of the graph of the function at the given value. f(x)=2x2+5x−7x2f ( x ) = 2 x ^ { 2 } + 5 x - \frac { 7 } { x ^ { 2 } }f(x)=2x2+5x−x27​ when x=−4x = - 4x=−4

Find dydx by implicit differentiation. \text { Find } \frac { d y } { d x } \text { by implicit differentiation. } Find dxdy​ by implicit differentiation. x2+y2=25x ^ { 2 } + y ^ { 2 } = 25x2+y2=25

Newton's Method to approximate the zero(s) of the function f(x)=x5+4x+1f ( x ) = x ^ { 5 } + 4 x + 1f(x)=x5+4x+1 accurate to three decimal places.

Find the second derivative of the function f(x)=8x59. \text { Find the second derivative of the function } f ( x ) = 8 x ^ { \frac { 5 } { 9 } } \text {. } Find the second derivative of the function f(x)=8x95​.

Find the second derivative of the function f(x)=x4sec⁡x. \text { Find the second derivative of the function } f ( x ) = x ^ { 4 } \sec x \text {. } Find the second derivative of the function f(x)=x4secx.

Find the derivative of the function sin⁡x2x+cos⁡x. Simplify your answer. \text { Find the derivative of the function } \frac { \sin x } { 2 x + \cos x } \text {. Simplify your answer. } Find the derivative of the function 2x+cosxsinx​. Simplify your answer.

Find dydx by implicit differentiation. \text { Find } \frac { d y } { d x } \text { by implicit differentiation. } Find dxdy​ by implicit differentiation. x2+5x+9xy−y2=4x ^ { 2 } + 5 x + 9 x y - y ^ { 2 } = 4x2+5x+9xy−y2=4

Graphs and Models

A Preview of Calculus

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

$\text { Find } f ^ { tt} ( x ) \text { if } f ( x ) = ( 7 + 4 x ) e ^ { - 4 x }$

$\text { Find the derivative of the function } f ( x ) = \sqrt { 7 x - 3 } \text { using the limiting process. }$

$\text { Differentiate the function } f ( x ) = \ln \left( \frac { e ^ { 5 x } + 1 } { e ^ { 2 x } + 1 } \right) \text {. }$

$\text { Use logarithmic differentiation to find } \frac { d y } { d x } \text {. }$ $y = \frac { 5 x - 3 } { ( 3 x + 2 ) ^ { 3 } }$

$\text { Find the derivative of the function } f ( x ) = \frac { x ^ { 5 } - 9 } { x ^ { 4 } } \text {. }$

$\text { Use the Quotient Rule to differentiate the function } f ( x ) = \frac { \sin x } { x ^ { 2 } + 3 } \text {. }$

$\text { Use implicit differentiation to find } \frac { d y } { d x } \text {. }$ $7 x ^ { 2 } + 8 \ln x y = 14$

Find the slope $m \text { of the line tangent to the graph of the function } f ( x ) = 2 - 7 x \text { at the }$ point $( - 1,9 )$

$\text { Find the derivative of the function } f ( x ) = 10 \arcsin ( x - 9 ) \text {. }$

Find the slope of the graph of the function at the given value. $f ( x ) = 2 x ^ { 2 } + 5 x - \frac { 7 } { x ^ { 2 } }$ when $x = - 4$

$\text { Find } \frac { d y } { d x } \text { by implicit differentiation. }$ $x ^ { 2 } + y ^ { 2 } = 25$

Newton's Method to approximate the zero(s) of the function $f ( x ) = x ^ { 5 } + 4 x + 1$ accurate to three decimal places.

$\text { Find the second derivative of the function } f ( x ) = 8 x ^ { \frac { 5 } { 9 } } \text {. }$

$\text { Find the second derivative of the function } f ( x ) = x ^ { 4 } \sec x \text {. }$

$\text { Find the derivative of the function } \frac { \sin x } { 2 x + \cos x } \text {. Simplify your answer. }$

$\text { Find } \frac { d y } { d x } \text { by implicit differentiation. }$ $x ^ { 2 } + 5 x + 9 x y - y ^ { 2 } = 4$