Deck 2: Derivatives

Question 1

Suppose that $h ( t )$ represents the height, in feet, of a person $t$ years old. In real world terms, what does $h ( 10 )$ represent? What is its unit? What does $h ^ { \prime } ( t )$ represents and what is its unit?

Answer 1

$h ( 10 )$ represents the height of a 10-year old person. Its unit is feet.
$h ^ { \prime } ( t )$ represents the rate of change of the height when a person is t years old. Its unit is feet per year.

Question 2

The function $f ( x ) = 9 - 2 x + x ^ { 2 }$ is both continuous and differentiable at $x = 0$ Write these facts as limit statements.

Answer 2

Since f is continuous at x = 0, $\lim _ { x \rightarrow 0 } \left( 9 - 2 x + x ^ { 2 } \right) = f ( 0 ) = 9$
Since f is differentiable at x = 0, $\lim _ { h \rightarrow 0 } \frac { - 2 h + 2 x h + h ^ { 2 } } { h }$ exists.
Note that there are alternate ways of writing the answer

Question 3

Suppose $f ( 1 ) = 2 , \lim _ { x \rightarrow 1 ^ { - } } f ( x ) = 2 , \text { and } \lim _ { x \rightarrow 1 ^ { + } } f ( x ) = 2 , \lim _ { x \rightarrow 1 ^ { - } } \frac { f ( x ) - f ( 1 ) } { x - 1 } = - 2$ and $\lim _ { x \rightarrow 1 ^ { + } } \frac { f ( x ) - f ( 1 ) } { x - 1 } = 1$ Is $f$ continuous and/or differentiable at $x = 1 ?$

A) f is not continuous but differentiable at x = 1
B) f is neither continuous nor differentiable at x = 1
C) f is continuous at but not differentiable at x = 1
D) f is both continuous and differentiable at x = 1

Answer 3

f is continuous at but not differentiable at x = 1

Question 4

Suppose $f ( 1 ) = 2 , \lim _ { x \rightarrow 1 ^ { - } } f ( x ) = 2 , \text { and } \lim _ { x \rightarrow 1 ^ { + } } f ( x ) = 2 , \lim _ { h \rightarrow 0 ^ { - } } \frac { f ( 1 + h ) - f ( 1 ) } { h } = - 2$ and $\lim _ { h \rightarrow 0 ^ { + } } \frac { f ( 1 + h ) - f ( 1 ) } { h } = - 2$ Is $f$ continuous and/or differentiable at $x = 1 ?$

A) f is not continuous but differentiable at x = 1
B) f is neither continuous nor differentiable at x = 1
C) f is continuous but not differentiable at x = 1
D) f is both continuous and differentiable at x = 1

Question 5

Use the definition of derivative: $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ to find $f ^ { \prime } ( - 1 ) \text {, if } f ( x ) = x ^ { 2 }$

Question 6

Use the definition of derivative: $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ to find $f ^ { \prime } ( - 2 ) \text {, if } f ( x ) = \frac { 2 } { x }$

Question 7

Use the definition of derivative: $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ to find $f ^ { \prime } ( 1 ) \text {, if } f ( x ) = \frac { x + 1 } { x - 2 }$

Question 8

Use the definition of derivative: $\lim _ { z \rightarrow c } \frac { f ( z ) - f ( c ) } { z - c }$ to find $f ^ { \prime } ( - 1 ) \text {, if } f ( x ) = x ^ { 2 }$

Question 9

Use the definition of derivative: $\lim _ { z \rightarrow c } \frac { f ( z ) - f ( c ) } { z - c }$ to find $f ^ { \prime } ( - 2 ) \text {, if } f ( x ) = \frac { 2 } { x }$

Question 10

Use the definition of derivative: $\lim _ { z \rightarrow c } \frac { f ( z ) - f ( c ) } { z - c }$ to find $f ^ { \prime } ( 1 ) \text {, if } f ( x ) = \frac { x + 1 } { x - 2 }$

Question 11

Use the definition of derivative: $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ to find $f ^ { \prime } ( x ) , \text { if } f ( x ) = \frac { 3 } { x + 1 }$

Question 12

Use the definition of derivative: $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ to find $f ^ { \prime } ( x ) , \text { if } f ( x ) = 2 \sqrt { x }$

Question 13

Use the definition of derivative: $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ to find $f ^ { \prime } ( x ) \text {, if } f ( x ) = \frac { 1 } { x ^ { 2 } }$

Question 14

Use the definition of derivative: $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ to find $f ^ { \prime } ( x ) , \text { if } f ( x ) = \sqrt { x + 1 }$

Question 15

Given $f ( x ) = \left\{ \begin{array} { l } - 2 x \text { if } x < 1 \\x - 3 \text { if } x \geq 1\end{array} \right.$ , is $f$ continuous and/or differentiable at $x = 1 ?$ Explain.

A) f is not continuous but differentiable at x = 1
B) f is neither continuous nor differentiable at x = 1
C) f is continuous but not differentiable at x = 1
D) f is both continuous and differentiable at x = 1

Question 16

Given $f ( x ) = \left\{ \begin{array} { l } x ^ { 2 } - 2 \text { if } x < 2 \\2 x + 1 \text { if } x \geq 2\end{array} \right.$ , is $f$ continuous and/or differentiable at $x = 2 ?$ Explain.

A) f is continuous but not differentiable at x = 2
B) f is differentiable but not continuous at x = 2
C) f is neither continuous nor differentiable at x = 2
D) f is both continuous and differentiable at x = 2

Question 17

Use the Intermediate Value Theorem to show that $f ( x ) = x ^ { 2 } - 2$ has at least one zero on [0, 2].

Question 18

Use the Intermediate Value Theorem to show that $f ( x ) = x ^ { 3 } + 2$ has at least one zero on [- 2, 1].

Question 19

Suppose $f$ is a piecewise-defined function, equal to $g ( x ) \text { if } x < 3 \text {, and } h ( x ) \text { if } x \geq 3 \text {, }$ where $g \text { and } h$ are continuous and differentiable everywhere. If $g ^ { \prime } ( 3 ) = h ^ { \prime } ( 3 )$ is the function $f$ differentiable at $x = 3 ?$ Explain why or why not.

Question 20

Suppose $f , g \text { and } h$ are functions with values $f ( 1 ) = - 3 , g ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 0 , g ^ { \prime } ( 1 ) = 3 \text {. }$ Find $( f g ) ^ { \prime } ( 1 )$

A) 7
B) 9
C) - 9
D) - 7

Question 21

Suppose $f , g \text { and } h$ are functions with values $f ( 1 ) = - 3 , g ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 0 , g ^ { \prime } ( 1 ) = 3 \text {. }$ Find $( 2 f - 5 g )^{\prime} ( 1 )$

A) 15
B) - 12
C) 12
D) - 15

Question 22

Suppose $f , g \text { and } h$ are functions with values $f ( 1 ) = - 3 , g ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 0 , g ^ { \prime } ( 1 ) = 3 \text {. }$ Find $\left( \frac { g } { f } \right) ^ { \prime } ( 1 )$

A) 3
B) $- \frac { 9 } { 4 }$
C) - 1
D) 1

Question 23

Find constants $a \text { and } b$ so that $f ( x ) = \left\{ \begin{array} { l l } a x + b & \text { if } x < 1 \\b x ^ { 2 } + 1 & \text { if } x \geq 1\end{array} \right.$ , is continuous and differentiable everywhere?

A) $a = 2 , b = 1$
B) $a = 1 , b = 1$
C) $a = 1 , b = \frac { - 1 } { 2 }$
D) $a = 1 , b = \frac { 1 } { 2 }$

Question 24

$\begin{array} { | l | l | l | l | l | } \hline \mathrm { X } & \mathrm { g } ( \mathrm { x } ) & \mathrm { h } ( \mathrm { x } ) & \mathrm { g } ^ { \prime } ( \mathrm { x } ) & \mathrm { h } ^ { \prime } ( \mathrm { x } ) \\\hline - 3 & 0 & 3 & 1 & 0 \\\hline - 2 & 1 & 2 & 2 & - 3 \\\hline - 1 & 3 & 0 & - 1 & - 2 \\\hline 0 & 2 & 3 & - 2 & 3 \\\hline 1 & 0 & - 1 & - 2 & - 2 \\\hline 2 & - 2 & - 2 & - 1 & 0 \\\hline 3 & - 3 & 0 & 0 & 1 \\\hline\end{array}$

-Use the table above to find $f ^ { \prime } ( 3 )$ if $f ( x ) = 5 g ( x ) - 4 h ( x )$

A) 4
B) - 1
C) 1
D) - 4

Question 25

$\begin{array} { | l | l | l | l | l | } \hline \mathrm { X } & \mathrm { g } ( \mathrm { x } ) & \mathrm { h } ( \mathrm { x } ) & \mathrm { g } ^ { \prime } ( \mathrm { x } ) & \mathrm { h } ^ { \prime } ( \mathrm { x } ) \\\hline - 3 & 0 & 3 & 1 & 0 \\\hline - 2 & 1 & 2 & 2 & - 3 \\\hline - 1 & 3 & 0 & - 1 & - 2 \\\hline 0 & 2 & 3 & - 2 & 3 \\\hline 1 & 0 & - 1 & - 2 & - 2 \\\hline 2 & - 2 & - 2 & - 1 & 0 \\\hline 3 & - 3 & 0 & 0 & 1 \\\hline\end{array}$

-Use the table above to find $f ^ { \prime } ( 2 )$ if $f ( x ) = \frac { 2 g ( x ) } { h ( x ) }$

A) 4
B) - 1
C) 1
D) - 4

Question 26

$\begin{array} { | l | l | l | l | l | } \hline \mathrm { X } & \mathrm { g } ( \mathrm { x } ) & \mathrm { h } ( \mathrm { x } ) & \mathrm { g } ^ { \prime } ( \mathrm { x } ) & \mathrm { h } ^ { \prime } ( \mathrm { x } ) \\\hline - 3 & 0 & 3 & 1 & 0 \\\hline - 2 & 1 & 2 & 2 & - 3 \\\hline - 1 & 3 & 0 & - 1 & - 2 \\\hline 0 & 2 & 3 & - 2 & 3 \\\hline 1 & 0 & - 1 & - 2 & - 2 \\\hline 2 & - 2 & - 2 & - 1 & 0 \\\hline 3 & - 3 & 0 & 0 & 1 \\\hline\end{array}$

-Use the table above to find $f ^ { \prime } ( - 1 )$ if $f ( x ) = \frac { g ( x ) h ( x ) + 1 } { g ( x ) }$

A) 19/9
B) - 19/9
C) 17/9
D) - 17/9

Question 27

$\begin{array} { | l | l | l | l | l | } \hline \mathrm { X } & \mathrm { g } ( \mathrm { x } ) & \mathrm { h } ( \mathrm { x } ) & \mathrm { g } ^ { \prime } ( \mathrm { x } ) & \mathrm { h } ^ { \prime } ( \mathrm { x } ) \\\hline - 3 & 0 & 3 & 1 & 0 \\\hline - 2 & 1 & 2 & 2 & - 3 \\\hline - 1 & 3 & 0 & - 1 & - 2 \\\hline 0 & 2 & 3 & - 2 & 3 \\\hline 1 & 0 & - 1 & - 2 & - 2 \\\hline 2 & - 2 & - 2 & - 1 & 0 \\\hline 3 & - 3 & 0 & 0 & 1 \\\hline\end{array}$

-Use the table above to find $f ^ { \prime } ( - 2 )$ if $f ( x ) = g ( h ( x ) )$

A) - 3
B) 3
C) - 1
D) 1

Question 28

$\begin{array} { | l | l | l | l | l | } \hline \mathrm { X } & \mathrm { g } ( \mathrm { x } ) & \mathrm { h } ( \mathrm { x } ) & \mathrm { g } ^ { \prime } ( \mathrm { x } ) & \mathrm { h } ^ { \prime } ( \mathrm { x } ) \\\hline - 3 & 0 & 3 & 1 & 0 \\\hline - 2 & 1 & 2 & 2 & - 3 \\\hline - 1 & 3 & 0 & - 1 & - 2 \\\hline 0 & 2 & 3 & - 2 & 3 \\\hline 1 & 0 & - 1 & - 2 & - 2 \\\hline 2 & - 2 & - 2 & - 1 & 0 \\\hline 3 & - 3 & 0 & 0 & 1 \\\hline\end{array}$

-Use the table above to find $f ^ { \prime } ( - 2 )$ if $f ( x ) = h ( g ( x ) )$

A) 4
B) - 1
C) 1
D) - 4

Question 29

$\begin{array} { | l | l | l | l | l | } \hline \mathrm { X } & \mathrm { g } ( \mathrm { x } ) & \mathrm { h } ( \mathrm { x } ) & \mathrm { g } ^ { \prime } ( \mathrm { x } ) & \mathrm { h } ^ { \prime } ( \mathrm { x } ) \\\hline - 3 & 0 & 3 & 1 & 0 \\\hline - 2 & 1 & 2 & 2 & - 3 \\\hline - 1 & 3 & 0 & - 1 & - 2 \\\hline 0 & 2 & 3 & - 2 & 3 \\\hline 1 & 0 & - 1 & - 2 & - 2 \\\hline 2 & - 2 & - 2 & - 1 & 0 \\\hline 3 & - 3 & 0 & 0 & 1 \\\hline\end{array}$

-Use the table above to find $f ^ { \prime } ( 1 )$ if $f ( x ) = ( h ( x ) ) ^ { 3 }$

A) 3
B) - 3
C) 6
D) - 6

Question 30

$\begin{array} { | l | l | l | l | l | } \hline \mathrm { X } & \mathrm { g } ( \mathrm { x } ) & \mathrm { h } ( \mathrm { x } ) & \mathrm { g } ^ { \prime } ( \mathrm { x } ) & \mathrm { h } ^ { \prime } ( \mathrm { x } ) \\\hline - 3 & 0 & 3 & 1 & 0 \\\hline - 2 & 1 & 2 & 2 & - 3 \\\hline - 1 & 3 & 0 & - 1 & - 2 \\\hline 0 & 2 & 3 & - 2 & 3 \\\hline 1 & 0 & - 1 & - 2 & - 2 \\\hline 2 & - 2 & - 2 & - 1 & 0 \\\hline 3 & - 3 & 0 & 0 & 1 \\\hline\end{array}$

-Use the table above to find $f ^ { \prime } ( - 1 )$ if $f ( x ) = \sqrt { g ( x ) }$

A) $\frac { \sqrt { 3 } } { 2 }$
B) $\frac { 2 } { \sqrt { 3 } }$
C) $\frac { 1 } { 2 \sqrt { 3 } }$
D) $\frac { - 1 } { 2 \sqrt { 3 } }$

Question 31

$\begin{array} { | l | l | l | l | l | } \hline \mathrm { X } & \mathrm { g } ( \mathrm { x } ) & \mathrm { h } ( \mathrm { x } ) & \mathrm { g } ^ { \prime } ( \mathrm { x } ) & \mathrm { h } ^ { \prime } ( \mathrm { x } ) \\\hline - 3 & 0 & 3 & 1 & 0 \\\hline - 2 & 1 & 2 & 2 & - 3 \\\hline - 1 & 3 & 0 & - 1 & - 2 \\\hline 0 & 2 & 3 & - 2 & 3 \\\hline 1 & 0 & - 1 & - 2 & - 2 \\\hline 2 & - 2 & - 2 & - 1 & 0 \\\hline 3 & - 3 & 0 & 0 & 1 \\\hline\end{array}$

-Use the table above to find $f ^ { \prime } ( - 1 )$ if $f ( x ) = h \left( x ^ { 2 } g ( x ) \right)$

A) 1
B) - 1
C) 7
D) - 7

Question 32

$\begin{array} { | l | l | l | l | l | } \hline \mathrm { X } & \mathrm { g } ( \mathrm { x } ) & \mathrm { h } ( \mathrm { x } ) & \mathrm { g } ^ { \prime } ( \mathrm { x } ) & \mathrm { h } ^ { \prime } ( \mathrm { x } ) \\\hline - 3 & 0 & 3 & 1 & 0 \\\hline - 2 & 1 & 2 & 2 & - 3 \\\hline - 1 & 3 & 0 & - 1 & - 2 \\\hline 0 & 2 & 3 & - 2 & 3 \\\hline 1 & 0 & - 1 & - 2 & - 2 \\\hline 2 & - 2 & - 2 & - 1 & 0 \\\hline 3 & - 3 & 0 & 0 & 1 \\\hline\end{array}$

-Use the table above to find $f ^ { \prime } ( 1 )$ if $f ( x ) = h \left( 3 - 2 x ^ { 2 } \right)$

A) 2
B) - 2
C) 8
D) - 8

Question 33

Find the derivative of $f ( x ) = 5 - 2 e + 3 x - 2 x ^ { 5 } + \frac { 1 } { \sqrt [ 3 ] { x } }$

A) $- 1 - 10 x ^ { 4 } - \frac { 1 } { 3 } x ^ { - \frac { 1 } { 3 } }$
B) $3 - 10 x ^ { 4 } - \frac { 1 } { 3 } x ^ { - \frac { 1 } { 3 } }$
C) $3 - 10 x ^ { 4 } - \frac { 1 } { 3 } x ^ { - \frac { 4 } { 3 } }$
D) $3 - 10 x ^ { 4 } - \frac { 1 } { 3 } x ^ { - \frac { 2 } { 3 } }$

Question 34

Find the derivative of $f ( x ) = - 2 x ^ { 3 } + 3 x + 4 \sqrt { x } + 5$

A) $6 x ^ { 2 } + 3 + 2 x ^ { \frac { 1 } { 2 } }$
B) $- 6 x ^ { 2 } + 3 + 2 x ^ { \frac { 1 } { 2 } }$
C) $6 x ^ { 2 } + 3 + 2 x ^ { - \frac { 1 } { 2 } }$
D) $- 6 x ^ { 2 } + 3 + 2 x ^ { - \frac { 1 } { 2 } }$

Question 35

Find the derivative of $f ( x ) = \frac { x ^ { 2 } } { 1 - 2 x ^ { 3 } }$

A) $\frac { - 2 x ^ { 4 } + 2 x } { \left( 1 - 2 x ^ { 3 } \right) ^ { 2 } }$
B) $\frac { - 2 x ^ { 4 } - 2 x } { \left( 1 - 2 x ^ { 3 } \right) ^ { 2 } }$
C) $\frac { 2 x ^ { 4 } - 2 x } { \left( 1 - 2 x ^ { 3 } \right) ^ { 2 } }$
D) $\frac { 2 x ^ { 4 } + 2 x } { \left( 1 - 2 x ^ { 3 } \right) ^ { 2 } }$

Question 36

Find the derivative of $f ( x ) = \frac { 3 + 2 x } { 4 - 5 x }$

A) $\frac { - 23 } { ( 4 - 5 x ) ^ { 2 } }$
B) $\frac { 23 } { ( 4 - 5 x ) ^ { 2 } }$
C) $\frac { - 7 } { ( 4 - 5 x ) ^ { 2 } }$
D) $\frac { 7 } { ( 4 - 5 x ) ^ { 2 } }$

Question 37

Find the derivative of $f ( x ) = ( \sqrt [ 3 ] { x } + 2 \sqrt { x } ) ^ { 3 }$

Question 38

Find the derivative of $f ( x ) = \frac { \sqrt [ 3 ] { x ^ { 5 } } - 3 x ^ { 5 } } { x ^ { 3 } }$

A) $\frac { 4 } { 3 } x ^ { - \frac { 3 } { 5 } } - 6 x$
B) $\frac { 4 } { 3 } x ^ { - \frac { 3 } { 7 } } - 6 x$
C) $- \frac { 4 } { 3 } x ^ { - \frac { 7 } { 3 } } - 6 x$
D) $\frac { 4 } { 3 } x ^ { - \frac { 7 } { 3 } } - 6 x$

Question 39

Find the derivative of $f ( x ) = \frac { ( x + 2 ) ^ { 2 } } { \left( x ^ { 2 } - 4 \right) ( x + 2 ) }$

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

Question 40

Find the derivative of $f ( x ) = | 2 x + 1 |$

Question 41

Find the derivative of $f ( x ) = | 1 - 3 x |$

Question 42

Find the derivative of $f ( x ) = \left\{ \begin{array} { c } 4 x - 1 \text { if } x < 2 \\2 x ^ { 2 } + 3 \text { if } x \geq 2\end{array} \right.$

Question 43

Find the derivative of $f ( x ) = \frac { 2 } { 4 - 5 x ^ { 4 } }$

A) $\frac { - 20 x ^ { 3 } } { \left( 4 - 5 x ^ { 4 } \right) ^ { 2 } }$
B) $\frac { 40 x ^ { 3 } } { \left( 4 - 5 x ^ { 4 } \right) ^ { 2 } }$
C) $\frac { 20 x ^ { 3 } } { \left( 4 - 5 x ^ { 4 } \right) ^ { 2 } }$
D) $\frac { - 40 x ^ { 3 } } { \left( 4 - 5 x ^ { 4 } \right) ^ { 2 } }$

Question 44

Differentiate $f ( x ) = \left( \frac { x ^ { 3 } - 1 } { \sqrt { x } } \right) ^ { 3 }$ in three ways: (a) with the chain rule, (b) with the quotient rule but not chain rule, (c) without the chain or quotient rules.

Question 45

Differentiate $f ( x ) = \left( \frac { 2 x ^ { 3 } + 1 } { \sqrt [ 3 ] { x } } \right) ^ { 3 }$ in three ways: (a) with the chain rule, (b) with the quotient rule but not chain rule, (c) without the chain or quotient rules.

Question 46

Find the derivative of $f ( x ) = ( \sqrt [ 3 ] { x } + 4 ) ^ { 6 }$

A) $\frac { 2 \left( x ^ { \frac { 1 } { 3 } } + 4 \right) ^ { 5 } } { x ^ { \frac { 1 } { 3 } } }$
B) $\frac { 2 \left( x ^ { \frac { 1 } { 3 } } + 4 \right) ^ { 5 } } { x ^ { \frac { 2 } { 3 } } }$
C) $2 \left( x ^ { \frac { 1 } { 3 } } + 4 \right) ^ { 5 }$
D) $\frac { 2 \left( x ^ { \frac { 1 } { 3 } } + 4 \right) ^ { 4 } } { x ^ { \frac { 2 } { 3 } } }$

Question 47

Find the derivative of $f ( x ) = 2 x \left( 4 x ^ { 2 } + 1 \right) ^ { 7 }$

A) $\left( 112 x ^ { 2 } + 2 \right) \left( 4 x ^ { 2 } + 1 \right) ^ { 6 }$
B) $\left( 120 x ^ { 2 } + 2 \right) \left( 4 x ^ { 2 } + 1 \right) ^ { 7 }$
C) $\left( 120 x ^ { 2 } + 2 \right) \left( 4 x ^ { 2 } + 1 \right) ^ { 6 }$
D) $112 x ^ { 2 } \left( 4 x ^ { 2 } + 1 \right) ^ { 6 }$

Question 48

Find the derivative of $f ( x ) = \frac { 2 x - 1 } { \sqrt { x ^ { 2 } + 2 } }$

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

Question 49

Find the derivative of $f ( x ) = \left( 3 x \sqrt { x ^ { 2 } + 1 } \right) ^ { - 3 }$

Question 50

Find the derivative of $f ( x ) = \frac { ( 1 - \sqrt { x } ) ^ { 2 } } { 2 x ^ { 3 } - 5 x + 1 }$

Question 51

Find the derivative of $f ( x ) = \sqrt { 5 - \sqrt { 2 x + 1 } }$

Question 52

Find the derivative of $f ( x ) = ( \sqrt [ 3 ] { x } - 3 x ) ^ { - 2 }$

A) $\frac { 18 x ^ { \frac { 1 } { 3 } } - 2 } { 3 x ^ { \frac { 2 } { 3 } } \left( x ^ { \frac { 1 } { 3 } } - 3 x \right) ^ { 3 } }$
B) $\frac { 18 x ^ { \frac { 1 } { 3 } } - 2 } { 3 x ^ { \frac { 1 } { 3 } } \left( x ^ { \frac { 1 } { 3 } } - 3 x \right) ^ { 3 } }$
C) $\frac { 18 x ^ { \frac { 2 } { 3 } } - 2 } { 3 x ^ { \frac { 2 } { 3 } } \left( x ^ { \frac { 1 } { 3 } } - 3 x \right) ^ { 3 } }$
D) $\frac { 18 x ^ { \frac { 2 } { 3 } } - 2 } { 2 x ^ { \frac { 2 } { 3 } } \left( x ^ { \frac { 1 } { 3 } } - 3 x \right) ^ { 3 } }$

Question 53

Find the derivative of $f ( x ) = \left( 1 - 2 x ^ { 3 } \right) ^ { 3 } \left( 4 x ^ { 2 } + 1 \right) ^ { 6 }$

A) $3 \left( 1 - 2 x ^ { 3 } \right) ^ { 2 } \left( 4 x ^ { 2 } + 1 \right) ^ { 4 } \left( 4 x ^ { 2 } + 16 x + 1 \right)$
B) $3 \left( 1 - 2 x ^ { 3 } \right) ^ { 2 } \left( 4 x ^ { 2 } + 1 \right) ^ { 4 } \left( 32 x ^ { 4 } + 4 x ^ { 2 } + 16 x + 1 \right)$
C) $3 \left( 1 - 2 x ^ { 3 } \right) ^ { 2 } \left( 4 x ^ { 2 } + 1 \right) ^ { 5 } \left( 4 x ^ { 2 } + 16 x + 1 \right)$
D) $6 x \left( 1 - 2 x ^ { 3 } \right) ^ { 2 } \left( 4 x ^ { 2 } + 1 \right) ^ { 5 } \left( - 28 x ^ { 3 } - 3 x + 8 \right)$

Question 54

Find the derivative of $f ( x ) = 2 \left( \left( x ^ { 2 } + 1 \right) ^ { 5 } - 4 x \right) ^ { - 5 / 2 }$

A) $\frac { 50 x \left( x ^ { 2 } + 1 \right) ^ { 4 } } { \left( \left( x ^ { 2 } + 1 \right) ^ { 5 } - 4 x \right) ^ { \frac { 5 } { 2 } } }$
B) $\frac { 20 - 50 x \left( x ^ { 2 } + 1 \right) ^ { 4 } } { \left( \left( x ^ { 2 } + 1 \right) ^ { 5 } - 4 x \right) ^ { \frac { 7 } { 2 } } }$
C) $\frac { 50 x \left( x ^ { 2 } + 1 \right) ^ { 4 } - 20 } { \left( \left( x ^ { 2 } + 1 \right) ^ { 5 } - 4 x \right) ^ { \frac { 7 } { 2 } } }$
D) $\frac { 20 - 50 x \left( x ^ { 2 } + 1 \right) ^ { 4 } } { \left( \left( x ^ { 2 } + 1 \right) ^ { 5 } - 4 x \right) ^ { \frac { 5 } { 2 } } }$

Question 55

If $f ( x ) = ( x \sqrt { x + 2 } ) ^ { - 2 }$ , find $f ^ { \prime \prime } ( x ) .$

Question 56

Use implicit differentiation to find $\frac { d y } { d x }$ if $x y ^ { 2 } + 2 x ^ { 3 } + y ^ { 2 } = 10$

A) $\frac { 6 x ^ { 2 } + y ^ { 2 } } { 2 x y + 2 y }$
B) $\frac { - 6 x ^ { 2 } + y ^ { 2 } } { 2 x y + 2 y }$
C) $\frac { 6 x ^ { 2 } - y ^ { 2 } } { 2 x y + 2 y }$
D) $\frac { - 6 x ^ { 2 } - y ^ { 2 } } { 2 x y + 2 y }$

Question 57

Use implicit differentiation to find $\frac { d y } { d x }$ if $5 x y + x ^ { 2 } y - y ^ { 2 } x = 10$

A) $\frac { 10 } { 5 x - x ^ { 2 } + 2 x y }$

B) $\frac { y ^ { 2 } + 5 y + 2 x y } { 5 x - x ^ { 2 } + 2 x y }$

C) $\frac { y ^ { 2 } - 5 y } { 5 x + x ^ { 2 } }$

D) $\frac { y ^ { 2 } - 2 x y - 5 y } { 5 x + x ^ { 2 } - 2 x y }$

Question 58

Use implicit differentiation to find $\frac { d y } { d x }$ if $\sqrt { 2 y + 1 } = 4 x y$

A) $\frac { 1 - 8 y \sqrt { 2 y + 1 } } { 8 x \sqrt { 2 y + 1 } }$
B) $\frac { 1 - 4 y \sqrt { 2 y + 1 } } { 2 x \sqrt { 2 y + 1 } }$
C) $\frac { 4 y \sqrt { 2 y + 1 } } { 1 - 4 x \sqrt { 2 y + 1 } }$
D) $\frac { 4 y \sqrt { 2 y + 1 } } { 1 + 2 x \sqrt { 2 y + 1 } }$

Question 59

Find the equation of the tangent lines to the circle $x ^ { 2 } + y ^ { 2 } = 4$ at the points with x-coordinate $x = 1$

Question 60

Find the equation of the tangent lines to the graph of $- x ^ { 2 } + 2 y ^ { 2 } + 3 x = 3$ at the points with x-coordinate $x = 2$

Question 61

Find the derivative of $f ( x ) = \frac { \left( x ^ { 3 } - 4 \right) ^ { 5 } ( 5 x - 2 ) } { ( x + 1 ) ^ { - 3 } \left( 2 + x ^ { 2 } \right) ^ { 4 } }$

Question 62

Find the derivative of $f ( x ) = \frac { \sqrt { 2 x + 1 } \left( x ^ { 2 } - 1 \right) ^ { 4 } } { ( x - 3 ) ^ { 3 } ( 2 + x ) }$

Question 63

Find the derivative of $f ( x ) = \frac { 1 } { 3 + e ^ { 2 x } }$

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

Question 64

Find the derivative of $f ( x ) = e ^ { 5 x } \ln \left( 2 x ^ { 2 } + 1 \right)$

A) $e ^ { 5 x } \ln \left( 2 x ^ { 2 } + 1 \right) + \frac { e ^ { 5 x } } { \ln \left( 2 x ^ { 2 } + 1 \right) }$
B) $e ^ { 5 x } \ln \left( 2 x ^ { 2 } + 1 \right) + \frac { e ^ { 5 x } } { 2 x ^ { 2 } + 1 }$
C) $5 e ^ { 5 x } \ln \left( 2 x ^ { 2 } + 1 \right) + \frac { e ^ { 5 x } } { 2 x ^ { 2 } + 1 }$
D) $5 e ^ { 5 x } \ln \left( 2 x ^ { 2 } + 1 \right) + \frac { 4 x e ^ { 5 x } } { 2 x ^ { 2 } + 1 }$

Question 65

Find the derivative of $f ( x ) = \frac { 3 + x ^ { 2 } } { e ^ { 2 x } }$

A) $\frac { 2 x ^ { 2 } - 2 x + 6 } { e ^ { 2 x } }$
B) $\frac { x ^ { 2 } - 2 x + 3 } { e ^ { 2 x } }$
C) $\frac { - 2 x ^ { 2 } + 2 x - 6 } { e ^ { 2 x } }$
D) $\frac { - x ^ { 2 } + 2 x - 3 } { e ^ { 2 x } }$

Question 66

Find the derivative of $f ( x ) = \sqrt [ 3 ] { \ln \left( 2 x ^ { 2 } + 1 \right) }$

A) $\frac { 1 } { \left( 6 x ^ { 2 } + 3 \right) \sqrt [ 3 ] { \ln \left( 2 x ^ { 2 } + 1 \right) } }$
B) $\frac { 4 x } { \left( 6 x ^ { 2 } + 3 \right) \sqrt [ 3 ] { \left[ \ln \left( 2 x ^ { 2 } + 1 \right) \right] ^ { 2 } } }$
C) $\frac { 4 x } { \left( 6 x ^ { 2 } + 3 \right) \sqrt [ 3 ] { \ln \left( 2 x ^ { 2 } + 1 \right) } }$
D) $\frac { 1 } { \left( 6 x ^ { 2 } + 3 \right) \sqrt [ 3 ] { \left[ \ln \left( 2 x ^ { 2 } + 1 \right) \right] ^ { 2 } } }$

Question 67

Find the derivative of $f ( x ) = e ^ { 5 x } \ln \left( x ^ { 2 } + 1 \right)$

Question 68

Find the derivative of $f ( x ) = x ^ { 3 } \ln \left( 2 x ^ { 2 } \right)$

A) $3 x ^ { 2 } \ln \left( 2 x ^ { 2 } \right) + \frac { 1 } { 2 } x$
B) $3 x ^ { 2 } \ln \left( 2 x ^ { 2 } \right) + 2 x$
C) $3 x ^ { 2 } \ln \left( 2 x ^ { 2 } \right) + 2 x ^ { 2 }$
D) $3 x \ln \left( 2 x ^ { 2 } \right) + 2 x ^ { 2 }$

Question 69

Find the derivative of $f ( x ) = x ^ { 2 } e ^ { 3 x }$

A) $2 x e ^ { 3 x } + x ^ { 2 } e ^ { 3 x }$
B) $2 x e ^ { 3 x } + 3 x ^ { 2 } e ^ { 3 x }$
C) $x ^ { 2 } e ^ { 3 x } + 3 x e ^ { 3 x }$
D) $2 e ^ { 3 x } + 3 x ^ { 2 } e ^ { 3 x }$

Question 70

Find the derivative of $f ( x ) = \ln \left( \frac { x ^ { 5 } } { x ^ { 3 } + 2 x + 1 } \right)$

A) $\frac { 5 } { x } + \frac { 3 x + 2 } { x ^ { 3 } + 2 x + 1 }$
B) $\frac { 5 } { x } + \frac { 3 x ^ { 2 } + 2 } { x ^ { 3 } + 2 x + 1 }$
C) $\frac { 5 } { x } - \frac { 3 x ^ { 2 } + 2 } { x ^ { 3 } + 2 x + 1 }$
D) $\frac { 5 } { x } - \frac { 3 x ^ { 2 } - 2 } { x ^ { 3 } + 2 x + 1 }$

Question 71

Find the derivative of $f ( x ) = 5 ^ { x } + \ln \sqrt { x }$

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

Question 72

Find the derivative of $f ( x ) = x e ^ { 3 x ^ { 2 } } + 5 x \ln \sqrt [ 3 ] { x }$

A) $6 x ^ { 2 } e ^ { 3 x ^ { 2 } } + \frac { 5 } { 3 }$
B) $6 x ^ { 2 } e ^ { 3 x ^ { 2 } } + 5 \ln \sqrt [ 3 ] { x } + \frac { 5 } { 3 }$
C) $6 x ^ { 2 } e ^ { 3 x ^ { 2 } } + e ^ { 3 x ^ { 2 } } + \frac { 5 } { 3 }$
D) $6 x ^ { 2 } e ^ { 3 x ^ { 2 } } + e ^ { 3 x ^ { 2 } } + 5 \ln \sqrt [ 3 ] { x } + \frac { 5 } { 3 }$

Question 73

Find the derivative of $f ( x ) = \left\{ \begin{array} { c l } \ln ( x + 3 ) & \text { if } x < 1 \\4 x ^ { 2 } - 1 & \text { if } x \geq 1\end{array} \right.$

A) $f ^ { \prime } ( x ) = \left\{ \begin{array} { c l } \frac { 1 } { x + 3 } & \text { if } x < 1 \\4 x ^ { 2 } + 1 & \text { if } x \geq 1\end{array} \right.$

B) $f ^ { \prime } ( x ) = \left\{ \begin{array} { c l } \frac { 1 } { x + 3 } & \text { if } x < 1 \\\text { DNE } & \text { if } x = 1 \\4 x & \text { if } x > 1\end{array} \right.$

C) $f ^ { \prime } ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { x + 3 } & \text { if } x < 1 \\8 x & \text { if } x \geq 1\end{array} \right.$

D) $f ^ { \prime } ( x ) = \left\{ \begin{array} { c l } \frac { 1 } { x + 3 } & \text { if } x < 1 \\\text { DNE } & \text { if } x = 1 \\8 x & \text { if } x > 1\end{array} \right.$

Question 74

Find the derivative of $f ( x ) = \frac { e ^ { 3 x } \left( x ^ { 3 } + 2 \right) ^ { 5 } } { x ^ { 2 } \left( 5 e ^ { 2 x } + 1 \right) }$

A) $f ^ { \prime } ( x ) = \frac { e ^ { 3 x } \left( x ^ { 3 } + 2 \right) ^ { 5 } } { x ^ { 2 } \left( 5 e ^ { 2 x } + 1 \right) } \left( 3 x + \frac { 5 } { x ^ { 3 } + 2 } - \frac { 2 } { x } - \frac { 10 e ^ { 2 x } } { 5 e ^ { 2 x } + 1 } \right)$
B) $f ^ { \prime } ( x ) = \frac { e ^ { 3 x } \left( x ^ { 3 } + 2 \right) ^ { 5 } } { x ^ { 2 } \left( 5 e ^ { 2 x } + 1 \right) } \left( 3 e ^ { 3 x } + \frac { 5 x ^ { 2 } } { x ^ { 3 } + 2 } - \frac { 2 } { x } - \frac { 10 e ^ { 2 x } } { 5 e ^ { 2 x } + 1 } \right)$
C) $f ^ { \prime } ( x ) = \frac { e ^ { 3 x } \left( x ^ { 3 } + 2 \right) ^ { 5 } } { x ^ { 2 } \left( 5 e ^ { 2 x } + 1 \right) } \left( 3 + \frac { 5 x ^ { 2 } } { x ^ { 3 } + 2 } - \frac { 2 } { x } - \frac { 10 e ^ { 2 x } } { 5 e ^ { 2 x } + 1 } \right)$
D) $f ^ { \prime } ( x ) = \frac { e ^ { 3 x } \left( x ^ { 3 } + 2 \right) ^ { 5 } } { x ^ { 2 } \left( 5 e ^ { 2 x } + 1 \right) } \left( 3 + \frac { 15 x ^ { 2 } } { x ^ { 3 } + 2 } - \frac { 2 } { x } - \frac { 10 e ^ { 2 x } } { 5 e ^ { 2 x } + 1 } \right)$

Question 75

Find the derivative of $f ( x ) = \left( \frac { x ^ { 2 } } { x - 1 } \right) ^ { x }$

A) $f ^ { \prime } ( x ) = \left( \frac { x ^ { 2 } } { x - 1 } \right) ^ { x } ( 2 \ln x + 1 - \ln ( x - 1 ) )$
B) $f ^ { \prime } ( x ) = \left( \frac { x ^ { 2 } } { x - 1 } \right) ^ { x } \left( 2 \ln x + 2 - \ln ( x - 1 ) - \frac { x } { x - 1 } \right)$
C) $f ^ { \prime } ( x ) = \left( \frac { x ^ { 2 } } { x - 1 } \right) ^ { x } \left( 2 \ln x - \ln ( x - 1 ) - \frac { x } { x - 1 } \right)$
D) $f ^ { \prime } ( x ) = \left( \frac { x ^ { 2 } } { x - 1 } \right) ^ { x } \left( 2 - \ln ( x - 1 ) - \frac { x } { x - 1 } \right)$

Question 76

Find the derivative of $f ( x ) = \sin ^ { 2 } x + 3 \cos ^ { 2 } x$

A) $2 \sin x + 6 \cos x$
B) $2 \sin 2 x$
C) $4 \sin 2 x$
D) $- 2 \sin 2 x$

Question 77

Find the derivative of $f ( x ) = 2 ^ { x } \tan x + 2 x$

A) $x 2 ^ { x - 1 } \tan x + 2 ^ { x } \sec ^ { 2 } x + 2$
B) $x 2 ^ { x - 1 } \tan x + 2 ^ { x } \sec x \tan x + 2$
C) $2 ^ { x } \ln 2 \tan x + 2 ^ { x } \sec ^ { 2 } x + 2$
D) $2 ^ { x } \ln 2 \tan x + 2 ^ { x } \sec x \tan x + 2$

Question 78

Find the derivative of $f ( x ) = \frac { x ^ { 2 } \ln \left( 3 x ^ { 2 } \right) } { \sin 3 x }$

Question 79

Find the derivative of $f ( x ) = 3 x ^ { 2 } \tan ^ { - 1 } x ^ { 3 }$

A) $6 x \tan ^ { - 1 } x ^ { 3 } + \frac { 3 x ^ { 2 } } { 1 + x ^ { 9 } }$
B) $6 x \tan ^ { - 1 } x ^ { 3 } + \frac { 3 x ^ { 2 } } { 1 + x ^ { 6 } }$
C) $6 x \tan ^ { - 1 } x ^ { 3 } + \frac { 9 x ^ { 2 } } { 1 + x ^ { 6 } }$
D) $6 x \tan ^ { - 1 } x ^ { 3 } + \frac { 9 x ^ { 4 } } { 1 + x ^ { 6 } }$

Question 80

Find the derivative of $f ( x ) = 2 x \sqrt { \sin 2 x \cos 2 x }$