Question 1

Use the alternative form of the derivative to find the derivative of the function $$f ( x ) = \frac { 3 } { x ^ { 2 } } \text { at } x = 2$$

Accepted Answer

A)  $f ^ { t } ( 2 ) = \frac { 3 } { 4 }$ 
B)  $f ^ { t } ( 2 ) = - \frac { 3 } { 4 }$ 
C)  $f ^ { t } ( 2 ) = \frac { 3 } { 8 }$ 
D)  $f ^ { t} ( 2 ) = - \frac { 3 } { 2 }$ 
E)  $f ^ { t } ( 2 ) = - \frac { 9 } { 16 }$ 
A)  $f ^ { t } ( 2 ) = \frac { 3 } { 4 }$ 
B)  $f ^ { t } ( 2 ) = - \frac { 3 } { 4 }$ 
C)  $f ^ { t } ( 2 ) = \frac { 3 } { 8 }$ 
D)  $f ^ { t} ( 2 ) = - \frac { 3 } { 2 }$ 
E)  $f ^ { t } ( 2 ) = - \frac { 9 } { 16 }$

Question 2

$\text { Differentiate } 3 \sin x \cos y = 1 \text { with respect to } t ( x \text { and } y \text { are functions of } t ) \text {. }$

Accepted Answer

A)  $\sin x \cos y \frac { d x } { d t } = \cos x \sin y \frac { d y } { d t }$ 
B)  $\cos x \cos y \frac { d x } { d t } = \sin x \sin y \frac { d y } { d t }$ 
C)  $x \sin y \frac { d x } { d t } = \sin x \cos y \frac { d y } { d t }$ 
D)  $\cos x \cos y \frac { d x } { d t } = 3 \sin x \sin y \frac { d y } { d t }$ 
E)  $3 \sin x \sin y \frac { d x } { d t } = \cos x \cos y \frac { d y } { d t }$ 
A)  $\sin x \cos y \frac { d x } { d t } = \cos x \sin y \frac { d y } { d t }$ 
B)  $\cos x \cos y \frac { d x } { d t } = \sin x \sin y \frac { d y } { d t }$ 
C)  $x \sin y \frac { d x } { d t } = \sin x \cos y \frac { d y } { d t }$ 
D)  $\cos x \cos y \frac { d x } { d t } = 3 \sin x \sin y \frac { d y } { d t }$ 
E)  $3 \sin x \sin y \frac { d x } { d t } = \cos x \cos y \frac { d y } { d t }$

Question 3

Suppose a 15-centimeter pendulum moves according to the equation $\theta = 0.6 \cos 8 t$ where $\theta$ is the angular displacement from the vertical in radians and $t$ is the time in seconds. Determine the rate of change of $\theta$ when $t = 7$ seconds. Round your answer to four decimal places.

Accepted Answer

A)  $2.5034$ radians per second 
B)  $3.6185$ radians per second 
C)  $0.3129$ radians per second 
D)  $3.1535$ radians per second 
E)  $4.1724$ radians per second 
A)  $2.5034$ radians per second 
B)  $3.6185$ radians per second 
C)  $0.3129$ radians per second 
D)  $3.1535$ radians per second 
E)  $4.1724$ radians per second

Question 4

Find the slope of the graph of the function at the given value. $f ( x ) = 4 ( 5 x + 6 ) ^ { 2 }$ at $x = 3$

Accepted Answer

A)  $f ^ { t } ( 3 ) = 840$ 
B)  $f ^ { t } ( 3 ) = 360$ 
C)  $f ^ { t } ( 3 ) = 42$ 
D)  $f ^ { t } ( 3 ) = 168$ 
E)  $f ^ { t } ( 3 ) = 210$ 
A)  $f ^ { t } ( 3 ) = 840$ 
B)  $f ^ { t } ( 3 ) = 360$ 
C)  $f ^ { t } ( 3 ) = 42$ 
D)  $f ^ { t } ( 3 ) = 168$ 
E)  $f ^ { t } ( 3 ) = 210$

Question 5

$\text { Find the slope } m \text { of the line tangent to the graph of the function } g ( x ) = 9 - x ^ { 2 } \text { at the }$ point $( 4 , - 7 )$

Accepted Answer

A)  $m = 4$ 
B)  $m = 9$ 
C)  $m = - 8$ 
D)  $m = - 7$ 
E)  $m = - 18$ 
A)  $m = 4$ 
B)  $m = 9$ 
C)  $m = - 8$ 
D)  $m = - 7$ 
E)  $m = - 18$

Question 6

$\text { Evaluate } \frac { d y } { d x } \text { for the equation } 7 x y = 21 \text { at the given point } ( - 3 , - 1 ) \text {. Round your }$ answer to two decimal places.

Accepted Answer

A)  $\frac { d y } { d x } = 7.00$ 
B)  $\frac { d y } { d x } = 63.00$ 
C)  $\frac { d y } { d x } = - 0.33$ 
D)  $\frac { d y } { d x } = - 63.00$ 
E)  $\frac { d y } { d x } = - 3.00$ 
A)  $\frac { d y } { d x } = 7.00$ 
B)  $\frac { d y } { d x } = 63.00$ 
C)  $\frac { d y } { d x } = - 0.33$ 
D)  $\frac { d y } { d x } = - 63.00$ 
E)  $\frac { d y } { d x } = - 3.00$

Question 7

$\text { If the annual rate of inflation averages } 8 \% \text { over the next } 10 \text { years, the approximate }$ cost $C$ of goods or services during any year in that decade is $C ( t ) = P ( 1.08 ) ^ { t }$ where $t$ is the time in years and $P$ is the present cost. The price of an oil change for your car is presently $\$ 24.35$. Estimate the price 3 years from now. Round your answer to two decimal places.

Accepted Answer

A)  $\$ 26.30$ 
B)  $\$ 30.67$ 
C)  $\$ 78.89$ 
D)  $\$ 25.61$ 
E)  $\$ 94.31$ 
A)  $\$ 26.30$ 
B)  $\$ 30.67$ 
C)  $\$ 78.89$ 
D)  $\$ 25.61$ 
E)  $\$ 94.31$

Question 8

$\text { Find } \frac { d ^ { 2 } y } { d x ^ { 2 } } \text { in terms of } x \text { and } y \text {. }\] \[5 - 2 x y = 7 x - 3 y$

Accepted Answer

A)  $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 4 ( 2 y - 7 ) } { ( 3 - 2 x ) ^ { 2 } }$ 
B) $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 4 ( 2 + 7 y ) } { ( 3 + 2 x ) ^ { 2 } }$ 
C) $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { - 124 } { ( 3 + 2 x ) ^ { 3 } }$ 
D) $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 44 } { ( 3 - 2 x ) ^ { 3 } }$ 
E) $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 4 ( 2 y + 7 ) } { ( 3 + 2 x ) ^ { 2 } }$ 
A)  $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 4 ( 2 y - 7 ) } { ( 3 - 2 x ) ^ { 2 } }$ 
B) $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 4 ( 2 + 7 y ) } { ( 3 + 2 x ) ^ { 2 } }$ 
C) $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { - 124 } { ( 3 + 2 x ) ^ { 3 } }$ 
D) $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 44 } { ( 3 - 2 x ) ^ { 3 } }$ 
E) $\frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 4 ( 2 y + 7 ) } { ( 3 + 2 x ) ^ { 2 } }$

Question 9

$$\text { Find the derivative of the function } f ( t ) = 15 t ^ { 3 } + 6 \sec ( t ) \text {. }$$

Accepted Answer

A)  $f ^ { t} ( t ) = 45 t ^ { 2 } + 6 \sec ( t ) \tan ( t )$ 
B)  $f ^ { t } ( t ) = 3 t ^ { 2 } + 6 \sec ^ { 2 } ( t )$ 
C)  $f ^ {t} ( t ) = 45 t ^ { 2 } + 6 \tan ( t )$ 
D)  $f ^ { t } ( t ) = 3 t ^ { 2 } + 6 \sec ( t ) \tan ( t )$ 
E)  $f ^ { t} ( t ) = 45 t ^ { 2 } - 6 \sec ( t ) \tan ( t )$ 
A)  $f ^ { t} ( t ) = 45 t ^ { 2 } + 6 \sec ( t ) \tan ( t )$ 
B)  $f ^ { t } ( t ) = 3 t ^ { 2 } + 6 \sec ^ { 2 } ( t )$ 
C)  $f ^ {t} ( t ) = 45 t ^ { 2 } + 6 \tan ( t )$ 
D)  $f ^ { t } ( t ) = 3 t ^ { 2 } + 6 \sec ( t ) \tan ( t )$ 
E)  $f ^ { t} ( t ) = 45 t ^ { 2 } - 6 \sec ( t ) \tan ( t )$

Question 10

$\text { Find the derivative of the function } y = 8 \sin 5 x \text {. }$

Accepted Answer

A)  $y ^ { t } = 40 \sin 5 x$ 
B)  $y ^ { t} = 40 \cos 5 x$ 
C)  $y ^ { t } = - 8 \sin 5 x$ 
D)  $y ^ {t } = - 40 \cos 5 x$ 
E)  $y ^ { t } = 8 \cos 5 x$ 
A)  $y ^ { t } = 40 \sin 5 x$ 
B)  $y ^ { t} = 40 \cos 5 x$ 
C)  $y ^ { t } = - 8 \sin 5 x$ 
D)  $y ^ {t } = - 40 \cos 5 x$ 
E)  $y ^ { t } = 8 \cos 5 x$

Question 11

Suppose the position function for a free-falling object on a certain planet is given by $s ( t ) = - 16 t ^ { 2 } + v _ { 0 } t + s _ { 0 }$. A silver coin is dropped from the top of a building that is 1372 feet tall. Determine the average velocity of the coin over the time interval $[ 3,4 ]$.

Accepted Answer

A)  $- 113 \mathrm { ft } / \mathrm { sec }$ 
B)  $80 \mathrm { ft } / \mathrm { sec }$ 
C)  $112 \mathrm { ft } / \mathrm { sec }$ 
D)  $- 112 \mathrm { ft } / \mathrm { sec }$ 
E)  $- 80 \mathrm { ft } / \mathrm { sec }$ 
A)  $- 113 \mathrm { ft } / \mathrm { sec }$ 
B)  $80 \mathrm { ft } / \mathrm { sec }$ 
C)  $112 \mathrm { ft } / \mathrm { sec }$ 
D)  $- 112 \mathrm { ft } / \mathrm { sec }$ 
E)  $- 80 \mathrm { ft } / \mathrm { sec }$

Question 12

$\text { Find the derivative of the following function } f ( x ) = - 3 x ^ { 2 } + 6 x - 8 \text { using the limiting }$ process.

Accepted Answer

A)  $f ^ { t } ( x ) = - 6 x + 6$ 
B)  $f ^ { t } ( x ) = - 3 x + 6$ 
C)  $f ^ { t} ( x ) = - 6 x + 6 x - 8$ 
D)  $f ^ { t } ( x ) = - 3 x - 6$ 
E)  $f ^ { t } ( x ) = - 6 x - 6$ 
A)  $f ^ { t } ( x ) = - 6 x + 6$ 
B)  $f ^ { t } ( x ) = - 3 x + 6$ 
C)  $f ^ { t} ( x ) = - 6 x + 6 x - 8$ 
D)  $f ^ { t } ( x ) = - 3 x - 6$ 
E)  $f ^ { t } ( x ) = - 6 x - 6$

Question 13

$\text { Find } \frac { d y } { d x } \text { if } y = x ^ { 5 } e ^ { x ^ { 9 } }$

Accepted Answer

A)  $\frac { d y } { d x } = 5 x ^ { 6 } e ^ { x ^ { 9 } - 1 }$ 
B) $\frac { d y } { d x } = e ^ { x ^ { 9 } } \left( \frac { x ^ { 6 } } { 6 } + \frac { x ^ { 10 } } { 10 } \right)$ 
C)  $\frac { d y } { d x } = e ^ { x ^ { 9 } } \left( 5 x ^ { 4 } + 9 x ^ { 13 } \right)$ 
D)  $\frac { d y } { d x } = e ^ { x ^ { 9 } } \left( x ^ { 5 } + x ^ { 9 } \right)$ 
E)  $\frac { d y } { d x } = 5 x ^ { 6 } e ^ { x ^ { 9 } }$ 
A)  $\frac { d y } { d x } = 5 x ^ { 6 } e ^ { x ^ { 9 } - 1 }$ 
B) $\frac { d y } { d x } = e ^ { x ^ { 9 } } \left( \frac { x ^ { 6 } } { 6 } + \frac { x ^ { 10 } } { 10 } \right)$ 
C)  $\frac { d y } { d x } = e ^ { x ^ { 9 } } \left( 5 x ^ { 4 } + 9 x ^ { 13 } \right)$ 
D)  $\frac { d y } { d x } = e ^ { x ^ { 9 } } \left( x ^ { 5 } + x ^ { 9 } \right)$ 
E)  $\frac { d y } { d x } = 5 x ^ { 6 } e ^ { x ^ { 9 } }$

Question 14

Find the derivative of the following function using the limiting process. $$f ( x ) = - 4 x ^ { 2 } + 5 x$$

Accepted Answer

A)  $- 4$ 
B)  $- 4 x + 5$ 
C)  $- 8 x - 5$ 
D)  $- 8 x$ 
E)  $- 8 x + 5$ 
A)  $- 4$ 
B)  $- 4 x + 5$ 
C)  $- 8 x - 5$ 
D)  $- 8 x$ 
E)  $- 8 x + 5$

Question 15

$\text { Find the derivative of the function } f ( x ) = - 5 x ^ { 3 } - 2 \sin ( x ) \text {. }$

Accepted Answer

A)  $f ^ { t } ( x ) = - 15 x ^ { 2 } + 2 \cos ( x )$ 
B)  $f ^ { t } ( x ) = - 10 x ^ { 2 } - 2 \cos ( x )$ 
C)  $f ^ {t } ( x ) = - 5 x ^ { 2 } - 2 \cos ( x )$ 
D)  $f ^ {t } ( x ) = - 5 x ^ { 2 } + 2 \cos ( x )$ 
E)  $f ^ { t } ( x ) = - 15 x ^ { 2 } - 2 \cos ( x )$ 
A)  $f ^ { t } ( x ) = - 15 x ^ { 2 } + 2 \cos ( x )$ 
B)  $f ^ { t } ( x ) = - 10 x ^ { 2 } - 2 \cos ( x )$ 
C)  $f ^ {t } ( x ) = - 5 x ^ { 2 } - 2 \cos ( x )$ 
D)  $f ^ {t } ( x ) = - 5 x ^ { 2 } + 2 \cos ( x )$ 
E)  $f ^ { t } ( x ) = - 15 x ^ { 2 } - 2 \cos ( x )$

Question 16

Find the derivative of the following function using the limiting process. $f ( x ) = \frac { 1 } { x ^ { 4 } }$

Accepted Answer

A)  $f ^ {t } ( x ) = \frac { 4 } { x ^ { 5 } }$ 
B)  $f ^ { t } ( x ) = - \frac { 4 } { x ^ { 3 } }$ 
C)  $f ^ { t } ( x ) = \frac { 4 } { x ^ { 3 } }$ 
D)  $f ^ {t } ( x ) = - \frac { 5 } { x ^ { 5 } }$ 
E)  $f ^ { t } ( x ) = - \frac { 4 } { x ^ { 5 } }$ 
A)  $f ^ {t } ( x ) = \frac { 4 } { x ^ { 5 } }$ 
B)  $f ^ { t } ( x ) = - \frac { 4 } { x ^ { 3 } }$ 
C)  $f ^ { t } ( x ) = \frac { 4 } { x ^ { 3 } }$ 
D)  $f ^ {t } ( x ) = - \frac { 5 } { x ^ { 5 } }$ 
E)  $f ^ { t } ( x ) = - \frac { 4 } { x ^ { 5 } }$

Question 17

$\text { Find an equation of the line that is tangent to the graph of the function } f ( x ) = 8 x ^ { 2 }$ and parallel to the line $16 x + y + 6 = 0$.

Accepted Answer

A)  $16 x + y + 8 = 0$ 
B)  $12 x - y + 6 = 0$ 
C)  $16 x - y + 8 = 0$ 
D)  $16 x + y + 6 = 0$ 
E)  $12 x + y + 6 = 0$ 
A)  $16 x + y + 8 = 0$ 
B)  $12 x - y + 6 = 0$ 
C)  $16 x - y + 8 = 0$ 
D)  $16 x + y + 6 = 0$ 
E)  $12 x + y + 6 = 0$

Question 18

A ladder feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of feet per second. How fast is the top of the ladder
Moving down the wall when its base is feet from the wall? Round your answer to two decimal
Places.

Accepted Answer

A)  $\quad 5.00 \mathrm { ft } / \mathrm { sec }$ 
B)  $- 5.33 \mathrm { ft } / \mathrm { sec }$ 
C)  $- 1.71 \mathrm { ft } / \mathrm { sec }$ 
D)  $\quad 5.33 \mathrm { ft } / \mathrm { sec }$ 
E)  $- 6.00 \mathrm { ft } / \mathrm { sec }$ 
A)  $\quad 5.00 \mathrm { ft } / \mathrm { sec }$ 
B)  $- 5.33 \mathrm { ft } / \mathrm { sec }$ 
C)  $- 1.71 \mathrm { ft } / \mathrm { sec }$ 
D)  $\quad 5.33 \mathrm { ft } / \mathrm { sec }$ 
E)  $- 6.00 \mathrm { ft } / \mathrm { sec }$

Question 19

$\text { Use implicit differentiation to find } \frac { d y } { d x } \text { at the point } ( 7,1 ) \text {. }$ $4 x y + 5 \ln y = 28$

Accepted Answer

A)  $\frac { 8 } { 11 }$ 
B)  $- \frac { 4 } { 33 }$ 
C)  $\frac { 4 } { 33 }$ 
D)  $- \frac { 8 } { 11 }$ 
E)  $\frac { 32 } { 33 }$ 
A)  $\frac { 8 } { 11 }$ 
B)  $- \frac { 4 } { 33 }$ 
C)  $\frac { 4 } { 33 }$ 
D)  $- \frac { 8 } { 11 }$ 
E)  $\frac { 32 } { 33 }$

Question 20

$\text { The radius, } r \text {, of a circle is decreasing at a rate of } 5 \text { centimeters per minute. }$ $\text { Find the rate of change of area, } A \text {, when the radius is } 6 .$

Accepted Answer

A)  $\frac { d A } { d t } = - 360 \pi \mathrm { sq } \mathrm { cm } / \mathrm { min }$ 
B)  $\frac { d A } { d t } = 360 \pi \mathrm { sq } \mathrm { cm } / \mathrm { min }$ 
C)  $\frac { d A } { d t } = - 60 \pi \mathrm { sq } \mathrm { cm } / \mathrm { min }$ 
D)  $\frac { d A } { d t } = 60 \pi \mathrm { sq } \mathrm { cm } / \mathrm { min }$ 
E)  $\frac { d A } { d t } = - 30 \pi \mathrm { sq } \mathrm { cm } / \mathrm { min }$ 
A)  $\frac { d A } { d t } = - 360 \pi \mathrm { sq } \mathrm { cm } / \mathrm { min }$ 
B)  $\frac { d A } { d t } = 360 \pi \mathrm { sq } \mathrm { cm } / \mathrm { min }$ 
C)  $\frac { d A } { d t } = - 60 \pi \mathrm { sq } \mathrm { cm } / \mathrm { min }$ 
D)  $\frac { d A } { d t } = 60 \pi \mathrm { sq } \mathrm { cm } / \mathrm { min }$ 
E)  $\frac { d A } { d t } = - 30 \pi \mathrm { sq } \mathrm { cm } / \mathrm { min }$

Use the alternative form of the derivative to find the derivative of the function $f ( x ) = \frac { 3 } { x ^ { 2 } } \text { at } x = 2$

$\text { Differentiate } 3 \sin x \cos y = 1 \text { with respect to } t ( x \text { and } y \text { are functions of } t ) \text {. }$

Suppose a 15-centimeter pendulum moves according to the equation $\theta = 0.6 \cos 8 t$ where $\theta$ is the angular displacement from the vertical in radians and $t$ is the time in seconds. Determine the rate of change of $\theta$ when $t = 7$ seconds. Round your answer to four decimal places.

Find the slope of the graph of the function at the given value. $f ( x ) = 4 ( 5 x + 6 ) ^ { 2 }$ at $x = 3$

$\text { Find the slope } m \text { of the line tangent to the graph of the function } g ( x ) = 9 - x ^ { 2 } \text { at the }$ point $( 4 , - 7 )$

$\text { Evaluate } \frac { d y } { d x } \text { for the equation } 7 x y = 21 \text { at the given point } ( - 3 , - 1 ) \text {. Round your }$ answer to two decimal places.

$\text { Find } \frac { d ^ { 2 } y } { d x ^ { 2 } } \text { in terms of } x \text { and } y \text {. } 5 - 2 x y = 7 x - 3 y$

$\text { Find the derivative of the function } f ( t ) = 15 t ^ { 3 } + 6 \sec ( t ) \text {. }$

$\text { Find the derivative of the function } y = 8 \sin 5 x \text {. }$

Suppose the position function for a free-falling object on a certain planet is given by $s ( t ) = - 16 t ^ { 2 } + v _ { 0 } t + s _ { 0 }$ . A silver coin is dropped from the top of a building that is 1372 feet tall. Determine the average velocity of the coin over the time interval $[ 3,4 ]$ .

$\text { Find the derivative of the following function } f ( x ) = - 3 x ^ { 2 } + 6 x - 8 \text { using the limiting }$ process.

$\text { Find } \frac { d y } { d x } \text { if } y = x ^ { 5 } e ^ { x ^ { 9 } }$

Find the derivative of the following function using the limiting process. $f ( x ) = - 4 x ^ { 2 } + 5 x$

$\text { Find the derivative of the function } f ( x ) = - 5 x ^ { 3 } - 2 \sin ( x ) \text {. }$

Find the derivative of the following function using the limiting process. $f ( x ) = \frac { 1 } { x ^ { 4 } }$

$\text { Find an equation of the line that is tangent to the graph of the function } f ( x ) = 8 x ^ { 2 }$ and parallel to the line $16 x + y + 6 = 0$ .

A ladder feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of feet per second. How fast is the top of the ladder Moving down the wall when its base is feet from the wall? Round your answer to two decimal Places.

$\text { Use implicit differentiation to find } \frac { d y } { d x } \text { at the point } ( 7,1 ) \text {. }$ $4 x y + 5 \ln y = 28$

$\text { The radius, } r \text {, of a circle is decreasing at a rate of } 5 \text { centimeters per minute. }$ $\text { Find the rate of change of area, } A \text {, when the radius is } 6 .$

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

Use the alternative form of the derivative to find the derivative of the function f(x)=3x2 at x=2f ( x ) = \frac { 3 } { x ^ { 2 } } \text { at } x = 2f(x)=x23​ at x=2

Differentiate 3sin⁡xcos⁡y=1 with respect to t(x and y are functions of t). \text { Differentiate } 3 \sin x \cos y = 1 \text { with respect to } t ( x \text { and } y \text { are functions of } t ) \text {. } Differentiate 3sinxcosy=1 with respect to t(x and y are functions of t).

Find the slope of the graph of the function at the given value. f(x)=4(5x+6)2f ( x ) = 4 ( 5 x + 6 ) ^ { 2 }f(x)=4(5x+6)2 at x=3x = 3x=3

Find d2ydx2 in terms of x and y. 5−2xy=7x−3y\text { Find } \frac { d ^ { 2 } y } { d x ^ { 2 } } \text { in terms of } x \text { and } y \text {. } 5 - 2 x y = 7 x - 3 y Find dx2d2y​ in terms of x and y. 5−2xy=7x−3y

Find the derivative of the function f(t)=15t3+6sec⁡(t). \text { Find the derivative of the function } f ( t ) = 15 t ^ { 3 } + 6 \sec ( t ) \text {. } Find the derivative of the function f(t)=15t3+6sec(t).

Find the derivative of the function y=8sin⁡5x. \text { Find the derivative of the function } y = 8 \sin 5 x \text {. } Find the derivative of the function y=8sin5x.

Find the derivative of the following function f(x)=−3x2+6x−8 using the limiting \text { Find the derivative of the following function } f ( x ) = - 3 x ^ { 2 } + 6 x - 8 \text { using the limiting } Find the derivative of the following function f(x)=−3x2+6x−8 using the limiting process.

Find dydx if y=x5ex9\text { Find } \frac { d y } { d x } \text { if } y = x ^ { 5 } e ^ { x ^ { 9 } } Find dxdy​ if y=x5ex9

Find the derivative of the following function using the limiting process. f(x)=−4x2+5xf ( x ) = - 4 x ^ { 2 } + 5 xf(x)=−4x2+5x

Find the derivative of the function f(x)=−5x3−2sin⁡(x). \text { Find the derivative of the function } f ( x ) = - 5 x ^ { 3 } - 2 \sin ( x ) \text {. } Find the derivative of the function f(x)=−5x3−2sin(x).

Find the derivative of the following function using the limiting process. f(x)=1x4f ( x ) = \frac { 1 } { x ^ { 4 } }f(x)=x41​

A ladder feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of feet per second. How fast is the top of the ladder Moving down the wall when its base is feet from the wall? Round your answer to two decimal Places.

Use implicit differentiation to find dydx at the point (7,1). \text { Use implicit differentiation to find } \frac { d y } { d x } \text { at the point } ( 7,1 ) \text {. } Use implicit differentiation to find dxdy​ at the point (7,1). 4xy+5ln⁡y=284 x y + 5 \ln y = 284xy+5lny=28

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

Use the alternative form of the derivative to find the derivative of the function $f ( x ) = \frac { 3 } { x ^ { 2 } } \text { at } x = 2$

$\text { Differentiate } 3 \sin x \cos y = 1 \text { with respect to } t ( x \text { and } y \text { are functions of } t ) \text {. }$

Find the slope of the graph of the function at the given value. $f ( x ) = 4 ( 5 x + 6 ) ^ { 2 }$ at $x = 3$

$\text { Find } \frac { d ^ { 2 } y } { d x ^ { 2 } } \text { in terms of } x \text { and } y \text {. } 5 - 2 x y = 7 x - 3 y$

$\text { Find the derivative of the function } f ( t ) = 15 t ^ { 3 } + 6 \sec ( t ) \text {. }$

$\text { Find the derivative of the function } y = 8 \sin 5 x \text {. }$

$\text { Find the derivative of the following function } f ( x ) = - 3 x ^ { 2 } + 6 x - 8 \text { using the limiting }$ process.

$\text { Find } \frac { d y } { d x } \text { if } y = x ^ { 5 } e ^ { x ^ { 9 } }$

Find the derivative of the following function using the limiting process. $f ( x ) = - 4 x ^ { 2 } + 5 x$

$\text { Find the derivative of the function } f ( x ) = - 5 x ^ { 3 } - 2 \sin ( x ) \text {. }$

Find the derivative of the following function using the limiting process. $f ( x ) = \frac { 1 } { x ^ { 4 } }$

$\text { Use implicit differentiation to find } \frac { d y } { d x } \text { at the point } ( 7,1 ) \text {. }$ $4 x y + 5 \ln y = 28$