Question 1

Find $\frac { d p } { d q }$ if $p = \ln \left( \frac { 7 q ^ { 2 } - 3 } { q } \right)$

Accepted Answer

A)  $\frac { 7 q ^ { 2 } - 3 } { q \left( 7 q ^ { 2 } - 3 \right) }$ 
B)  $\frac { 7 q ^ { 2 } + 3 } { q \left( 7 q ^ { 2 } - 3 \right) }$ 
C)  $\frac { 14 q ^ { 2 } + 3 } { q \left( 7 q ^ { 2 } - 3 \right) }$ 
D)  $\frac { 14 q ^ { 2 } - 3 } { q \left( 7 q ^ { 2 } - 3 \right) }$ 
E)  $\frac { 14 q ^ { 2 } + 7 } { q \left( 7 q ^ { 2 } - 3 \right) }$ 
A)  $\frac { 7 q ^ { 2 } - 3 } { q \left( 7 q ^ { 2 } - 3 \right) }$ 
B)  $\frac { 7 q ^ { 2 } + 3 } { q \left( 7 q ^ { 2 } - 3 \right) }$ 
C)  $\frac { 14 q ^ { 2 } + 3 } { q \left( 7 q ^ { 2 } - 3 \right) }$ 
D)  $\frac { 14 q ^ { 2 } - 3 } { q \left( 7 q ^ { 2 } - 3 \right) }$ 
E)  $\frac { 14 q ^ { 2 } + 7 } { q \left( 7 q ^ { 2 } - 3 \right) }$

Question 2

Use a graphing utility to graph the function $f ( x ) = 3 ^ { - x ^ { 2 } }$ .

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 3

Carbon-14(14C) dating assumes that the carbon on the Earth today has the same radioactive content as it did centuries ago. If this is true, then the amount of 14C absorbed by a tree that grew several centuries ago should be the same as the amount of 14C absorbed by a similar tree today. A piece of ancient charcoal contains only 18% as much of the radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal? (The half-life of 14C is 5715 years.) Round your answer to the nearest integer.

Accepted Answer

A) 2,310 years 
B) 33,123 years 
C) 2,315 years 
D) 14,139 years 
E) 14,144 years 
A) 2,310 years 
B) 33,123 years 
C) 2,315 years 
D) 14,139 years 
E) 14,144 years

Question 4

Use the properties of logarithms to expand $\ln \frac { 3 } { 10 }$

Accepted Answer

A)  $\ln 3 + \ln 10$ 
B)  $\ln 3 - \ln 10$ 
C)  $\frac { \ln 3 } { \ln 10 }$ 
D)  $( \ln 3 ) ( \ln 10 )$ 
E) none of the above 
A)  $\ln 3 + \ln 10$ 
B)  $\ln 3 - \ln 10$ 
C)  $\frac { \ln 3 } { \ln 10 }$ 
D)  $( \ln 3 ) ( \ln 10 )$ 
E) none of the above

Question 5

Find the derivative of $f ( x ) = x ^ { - 3 } - 10 e ^ { x }$

Accepted Answer

A)  $f ^ { \prime } ( x ) = 3 x ^ { 2 } - 10 e ^ { x }$ 
B)  $f ^ { \prime } ( x ) = - \frac { 3 } { x ^ { 4 } } - 10 x e ^ { x - 1 }$ 
C)  $f ^ { \prime } ( x ) = - \frac { 3 } { x ^ { 4 } } - 10 e ^ { x }$ 
D)  $f ^ { \prime } ( x ) = 3 x ^ { 2 } - 10 x e ^ { x - 1 }$ 
E)  $f ^ { \prime } ( x ) = - \frac { 3 } { x ^ { 4 } } - 10 x e ^ { x }$ 
A)  $f ^ { \prime } ( x ) = 3 x ^ { 2 } - 10 e ^ { x }$ 
B)  $f ^ { \prime } ( x ) = - \frac { 3 } { x ^ { 4 } } - 10 x e ^ { x - 1 }$ 
C)  $f ^ { \prime } ( x ) = - \frac { 3 } { x ^ { 4 } } - 10 e ^ { x }$ 
D)  $f ^ { \prime } ( x ) = 3 x ^ { 2 } - 10 x e ^ { x - 1 }$ 
E)  $f ^ { \prime } ( x ) = - \frac { 3 } { x ^ { 4 } } - 10 x e ^ { x }$

Question 6

To help their son buy a car on his 19th birthday, a boy's parents invest $1400 on his 10th birthday. If the investment pays an annual rate of 9% compounded continuously, how much is available on his 19th birthday?

Accepted Answer

A) $3118.94 
B) $3147.07 
C) $3040.65 
D) $2534.00 
E) $35,747.21 
A) $3118.94 
B) $3147.07 
C) $3040.65 
D) $2534.00 
E) $35,747.21

Question 7

What lump sum should be deposited in an account that will earn at an annual rate of 12%, compounded quarterly, to grow to $90,000 for retirement in 15 years?

Accepted Answer

A) $87,176.69 
B) $15,010.50 
C) $10,975.61 
D) $32,142.86 
E) $15,275.98 
A) $87,176.69 
B) $15,010.50 
C) $10,975.61 
D) $32,142.86 
E) $15,275.98

Question 8

Find the relative minima, and use a graphing utility to check your results. $y = 3 \ln x - 5 x$

Accepted Answer

A)  $- \frac { 5 } { 3 }$ 
B)  $- \frac { 3 } { 5 }$ 
C)  $- \frac { 1 } { 5 }$ 
D)  $0$ 
E) does not exist 
A)  $- \frac { 5 } { 3 }$ 
B)  $- \frac { 3 } { 5 }$ 
C)  $- \frac { 1 } { 5 }$ 
D)  $0$ 
E) does not exist

Question 9

A survey of high school seniors from a certain school district who took the SAT has determined that the mean score on the mathematics portion was 700 with a standard deviation of 13.5. By a normal probability density function the data can be modeled as $f ( x ) = \frac { 1 } { 13.5 \sqrt { 2 \pi } } e ^ { - ( x - 700 ) ^ { 2 } /3 64.5 }$ . Find the derivative of the model.

Accepted Answer

A)  $f ^ { \prime } ( x ) = \frac { - 2 \sqrt { 2 } ( x - 700 ) e ^ { - ( x -700 ) ^ { 2 } / 182.25 } } { 4,921 \sqrt { \pi } }$ 
B)  $f ^ { \prime } ( x ) = \frac { - 2 \sqrt { 2 } ( x - 700 ) e ^ { - ( x -700 ) ^ { 2 } / 364.5 } } { 9,842 \sqrt { \pi } }$ 
C)  $f ^ { \prime } ( x ) = \frac { \sqrt { 2 } ( x - 700 ) e ^ { - ( x - 700 ) ^ { 2 } / 364.5 } } { 4,921 \sqrt { \pi } }$ 
D)  $f ^ { \prime } ( x ) = \frac { \sqrt { 2 } ( x - 700 ) e ^ { - ( x - 700 ) ^ { 2 } / 364.5 } } { 9,842 \sqrt { \pi } }$ 
E)  $f ^ { \prime } ( x ) = \frac { \sqrt { 2 } ( x - 700 ) e ^ { - ( x - 70 ) ^ { 2 } / 182.25 } } { 4,921 \sqrt { \pi } }$ 
A)  $f ^ { \prime } ( x ) = \frac { - 2 \sqrt { 2 } ( x - 700 ) e ^ { - ( x -700 ) ^ { 2 } / 182.25 } } { 4,921 \sqrt { \pi } }$ 
B)  $f ^ { \prime } ( x ) = \frac { - 2 \sqrt { 2 } ( x - 700 ) e ^ { - ( x -700 ) ^ { 2 } / 364.5 } } { 9,842 \sqrt { \pi } }$ 
C)  $f ^ { \prime } ( x ) = \frac { \sqrt { 2 } ( x - 700 ) e ^ { - ( x - 700 ) ^ { 2 } / 364.5 } } { 4,921 \sqrt { \pi } }$ 
D)  $f ^ { \prime } ( x ) = \frac { \sqrt { 2 } ( x - 700 ) e ^ { - ( x - 700 ) ^ { 2 } / 364.5 } } { 9,842 \sqrt { \pi } }$ 
E)  $f ^ { \prime } ( x ) = \frac { \sqrt { 2 } ( x - 700 ) e ^ { - ( x - 70 ) ^ { 2 } / 182.25 } } { 4,921 \sqrt { \pi } }$

Question 10

Determine whether the function below has any horizontal asymptotes. $f ( x ) = \frac { e ^ { x } - e ^ { - x } } { 2 }$

Accepted Answer

A) horizontal asymptotes: y = 0 and y = 2 
B) no horizontal asymptotes 
C) horizontal asymptotes: y = 1 
D) horizontal asymptotes: y = 1 and y = 3 
E) horizontal asymptotes: y = 3 
A) horizontal asymptotes: y = 0 and y = 2 
B) no horizontal asymptotes 
C) horizontal asymptotes: y = 1 
D) horizontal asymptotes: y = 1 and y = 3 
E) horizontal asymptotes: y = 3

Question 11

Locate any relative extrema and inflection points of the function $y = x ^ { 6 } \ln \frac { x } { 6 }$ .

Accepted Answer

A) relative minimum at $x = 6 e ^ { \frac { - 1 } { 6 } }$ ; inflection point at $x = 6 e ^ { \frac { - 11 } { 30 } }$ 
B) relative minimum at $x = 6 e ^ { \frac { - 1 } { 6 } }$ ; no inflection points 
C) no relative maximum or minimum; inflection point at $x = 6 e ^ { \frac { - 1 } { 6 } }$ 
D) no relative extrema or inflection points. 
E) relative maximum at $x = 6 e ^ { \frac { - 1 } { 6 } }$ ; inflection point at $x = 6 e ^ { \frac { - 11 } { 30 } }$ 
A) relative minimum at $x = 6 e ^ { \frac { - 1 } { 6 } }$ ; inflection point at $x = 6 e ^ { \frac { - 11 } { 30 } }$ 
B) relative minimum at $x = 6 e ^ { \frac { - 1 } { 6 } }$ ; no inflection points 
C) no relative maximum or minimum; inflection point at $x = 6 e ^ { \frac { - 1 } { 6 } }$ 
D) no relative extrema or inflection points. 
E) relative maximum at $x = 6 e ^ { \frac { - 1 } { 6 } }$ ; inflection point at $x = 6 e ^ { \frac { - 11 } { 30 } }$

Question 12

Find an equation of the tangent line to the graph of $y = \log _ { 3 } x$ at the point $( 27,3 )$ .

Accepted Answer

A)  $y = 3 + \frac { 1 } { \ln 3 } ( x - 27 )$ 
B)  $y = 3 + \frac { 1 } { 3 \ln 27 } ( x - 27 )$ 
C)  $y = 3 + \frac { 1 } { 27 \ln 3 } ( x - 27 )$ 
D)  $y = 3 + \frac { 1 } { 27 } ( x - 27 )$ 
E) none of the above 
A)  $y = 3 + \frac { 1 } { \ln 3 } ( x - 27 )$ 
B)  $y = 3 + \frac { 1 } { 3 \ln 27 } ( x - 27 )$ 
C)  $y = 3 + \frac { 1 } { 27 \ln 3 } ( x - 27 )$ 
D)  $y = 3 + \frac { 1 } { 27 } ( x - 27 )$ 
E) none of the above

Question 13

Find the derivative of the following function. $y = \ln \left( 4 x ^ { 3 } - 9 x \right) - 3 x$

Accepted Answer

A)  $\frac { 12 x ^ { 2 } - 9 x } { x \left( 4 x ^ { 2 } - 9 \right) } - 3$ 
B)  $\frac { 1 } { x \left( 4 x ^ { 2 } - 9 \right) } - 3$ 
C)  $\frac { 12 x ^ { 2 } - 9 } { x \left( 4 x ^ { 2 } - 9 \right) } - 3$ 
D)  $\frac { 12 x ^ { 2 } } { x \left( 4 x ^ { 2 } - 9 \right) } - 3$ 
E)  $\frac { 1 } { x \left( 4 x ^ { 2 } - 1 \right) } - 3$ 
A)  $\frac { 12 x ^ { 2 } - 9 x } { x \left( 4 x ^ { 2 } - 9 \right) } - 3$ 
B)  $\frac { 1 } { x \left( 4 x ^ { 2 } - 9 \right) } - 3$ 
C)  $\frac { 12 x ^ { 2 } - 9 } { x \left( 4 x ^ { 2 } - 9 \right) } - 3$ 
D)  $\frac { 12 x ^ { 2 } } { x \left( 4 x ^ { 2 } - 9 \right) } - 3$ 
E)  $\frac { 1 } { x \left( 4 x ^ { 2 } - 1 \right) } - 3$

Question 14

What is the resulting balance if $5700 is invested for 6 years at an annual rate of 7% compounded monthly?

Accepted Answer

A)  $\$$ 5902.43 
B)  $\$$ 5936.86 
C)  $\$$ 8094.00 
D)  $\$$ 8664.60 
E)  $\$$ 11,811.92 
A)  $\$$ 5902.43 
B)  $\$$ 5936.86 
C)  $\$$ 8094.00 
D)  $\$$ 8664.60 
E)  $\$$ 11,811.92

Question 15

Use implicit differentiation to find $\frac { d y } { d x }$ . $9 e ^ { xy } + y ^ { 2 } = 15$

Accepted Answer

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

Question 16

Find the equation of the tangent line to $f ( x ) = 9 x + e ^ { x }$ at the point (0,1).

Accepted Answer

A)  $y = - 10 x - 10$ 
B)  $y = 10 x - 10$ 
C)  $y = 10 x - 1$ 
D)  $y = 10 x + 1$ 
E)  $y = - 10 x - 1$ 
A)  $y = - 10 x - 10$ 
B)  $y = 10 x - 10$ 
C)  $y = 10 x - 1$ 
D)  $y = 10 x + 1$ 
E)  $y = - 10 x - 1$

Question 17

Find $f ^ { \prime } ( t )$ if $f ( t ) = t ^ { 10 } 10 ^ { 3 t }$ .

Accepted Answer

A)  $f ^ { \prime } ( t ) = 10 t ^ { 9 } 10 ^ { 3 t } + 3 t ^ { 10 } 10 ^ { 3 t }$ 
B)  $f ^ { \prime } ( t ) = 10 t ^ { 9 } 10 ^ { 3 t } + 10 ^ { 3 t - 1 }$ 
C)  $f ^ { \prime } ( t ) = 10 t ^ { 9 } 10 ^ { 3 t } + 3 \ln ( 10 ) t ^ { 10 } 10 ^ { 3 t }$ 
D)  $f ^ { \prime } ( t ) = 10 t ^ { 9 } 10 ^ { 3 t } + 30 t ^ { 10 } 10 ^ { 3 t }$ 
E)  $f ^ { \prime } ( t ) = 10 t ^ { 9 } 30 ^ { 3 t - 1 }$ 
A)  $f ^ { \prime } ( t ) = 10 t ^ { 9 } 10 ^ { 3 t } + 3 t ^ { 10 } 10 ^ { 3 t }$ 
B)  $f ^ { \prime } ( t ) = 10 t ^ { 9 } 10 ^ { 3 t } + 10 ^ { 3 t - 1 }$ 
C)  $f ^ { \prime } ( t ) = 10 t ^ { 9 } 10 ^ { 3 t } + 3 \ln ( 10 ) t ^ { 10 } 10 ^ { 3 t }$ 
D)  $f ^ { \prime } ( t ) = 10 t ^ { 9 } 10 ^ { 3 t } + 30 t ^ { 10 } 10 ^ { 3 t }$ 
E)  $f ^ { \prime } ( t ) = 10 t ^ { 9 } 30 ^ { 3 t - 1 }$

Question 18

Evaluate the expression $256 ^ { \frac { 2 } { 4 } }$ .

Accepted Answer

A) 256 
B) 4 
C) 16 
D) 18 
E) 20 
A) 256 
B) 4 
C) 16 
D) 18 
E) 20

Question 19

Sketch the graph of the function $f ( x ) = e ^ { 3 x }$ .

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 20

Find $y ^ { \prime }$ . $y = \ln ( 5 x + 7 ) ^ { 1 / 8 }$

Accepted Answer

A)  $\frac { 5 } { 8 ( 5 x + 7 ) }$ 
B)  $\frac { 7 } { 5 ( 5 x + 7 ) }$ 
C)  $\frac { 7 } { 8 ( 5 x + 7 ) }$ 
D)  $\frac { 35 } { 8 ( 5 x + 7 ) }$ 
E)  $\frac { 40 } { 7 ( 5 x + 7 ) }$ 
A)  $\frac { 5 } { 8 ( 5 x + 7 ) }$ 
B)  $\frac { 7 } { 5 ( 5 x + 7 ) }$ 
C)  $\frac { 7 } { 8 ( 5 x + 7 ) }$ 
D)  $\frac { 35 } { 8 ( 5 x + 7 ) }$ 
E)  $\frac { 40 } { 7 ( 5 x + 7 ) }$

Find $\frac { d p } { d q }$ if $p = \ln \left( \frac { 7 q ^ { 2 } - 3 } { q } \right)$

Use a graphing utility to graph the function $f ( x ) = 3 ^ { - x ^ { 2 } }$ .

Use the properties of logarithms to expand $\ln \frac { 3 } { 10 }$

Find the derivative of $f ( x ) = x ^ { - 3 } - 10 e ^ { x }$

To help their son buy a car on his 19th birthday, a boy's parents invest $1400 on his 10th birthday. If the investment pays an annual rate of 9% compounded continuously, how much is available on his 19th birthday?

What lump sum should be deposited in an account that will earn at an annual rate of 12%, compounded quarterly, to grow to $90,000 for retirement in 15 years?

Find the relative minima, and use a graphing utility to check your results. $y = 3 \ln x - 5 x$

Determine whether the function below has any horizontal asymptotes. $f ( x ) = \frac { e ^ { x } - e ^ { - x } } { 2 }$

Locate any relative extrema and inflection points of the function $y = x ^ { 6 } \ln \frac { x } { 6 }$ .

Find an equation of the tangent line to the graph of $y = \log _ { 3 } x$ at the point $( 27,3 )$ .

Find the derivative of the following function. $y = \ln \left( 4 x ^ { 3 } - 9 x \right) - 3 x$

What is the resulting balance if $5700 is invested for 6 years at an annual rate of 7% compounded monthly?

Use implicit differentiation to find $\frac { d y } { d x }$ . $9 e ^ { xy } + y ^ { 2 } = 15$

Find the equation of the tangent line to $f ( x ) = 9 x + e ^ { x }$ at the point (0,1).

Find $f ^ { \prime } ( t )$ if $f ( t ) = t ^ { 10 } 10 ^ { 3 t }$ .

Evaluate the expression $256 ^ { \frac { 2 } { 4 } }$ .

Sketch the graph of the function $f ( x ) = e ^ { 3 x }$ .

Find $y ^ { \prime }$ . $y = \ln ( 5 x + 7 ) ^ { 1 / 8 }$

Fundamental Concepts of Algebra

Equations and Inequalities

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Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

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Further Applications of the Derivative

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Exam 11: Derivatives of Exponential and Logarithmic Functions

Find dpdq\frac { d p } { d q }dqdp​ if p=ln⁡(7q2−3q)p = \ln \left( \frac { 7 q ^ { 2 } - 3 } { q } \right)p=ln(q7q2−3​)

Use a graphing utility to graph the function f(x)=3−x2f ( x ) = 3 ^ { - x ^ { 2 } }f(x)=3−x2 .

Use the properties of logarithms to expand ln⁡310\ln \frac { 3 } { 10 }ln103​

Find the derivative of f(x)=x−3−10exf ( x ) = x ^ { - 3 } - 10 e ^ { x }f(x)=x−3−10ex

To help their son buy a car on his 19th birthday, a boy's parents invest $1400 on his 10th birthday. If the investment pays an annual rate of 9% compounded continuously, how much is available on his 19th birthday?

What lump sum should be deposited in an account that will earn at an annual rate of 12%, compounded quarterly, to grow to $90,000 for retirement in 15 years?

Find the relative minima, and use a graphing utility to check your results. y=3ln⁡x−5xy = 3 \ln x - 5 xy=3lnx−5x

Determine whether the function below has any horizontal asymptotes. f(x)=ex−e−x2f ( x ) = \frac { e ^ { x } - e ^ { - x } } { 2 }f(x)=2ex−e−x​

Locate any relative extrema and inflection points of the function y=x6ln⁡x6y = x ^ { 6 } \ln \frac { x } { 6 }y=x6ln6x​ .

Find an equation of the tangent line to the graph of y=log⁡3xy = \log _ { 3 } xy=log3​x at the point (27,3)( 27,3 )(27,3) .

Find the derivative of the following function. y=ln⁡(4x3−9x)−3xy = \ln \left( 4 x ^ { 3 } - 9 x \right) - 3 xy=ln(4x3−9x)−3x

What is the resulting balance if $5700 is invested for 6 years at an annual rate of 7% compounded monthly?

Use implicit differentiation to find dydx\frac { d y } { d x }dxdy​ . 9exy+y2=159 e ^ { xy } + y ^ { 2 } = 159exy+y2=15

Find the equation of the tangent line to f(x)=9x+exf ( x ) = 9 x + e ^ { x }f(x)=9x+ex at the point (0,1).

Find f′(t)f ^ { \prime } ( t )f′(t) if f(t)=t10103tf ( t ) = t ^ { 10 } 10 ^ { 3 t }f(t)=t10103t .

Evaluate the expression 25624256 ^ { \frac { 2 } { 4 } }25642​ .

Sketch the graph of the function f(x)=e3xf ( x ) = e ^ { 3 x }f(x)=e3x .

Find y′y ^ { \prime }y′ . y=ln⁡(5x+7)1/8y = \ln ( 5 x + 7 ) ^ { 1 / 8 }y=ln(5x+7)1/8

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Integration and Its Applications

Techniques of Integration

Functions of Several Variables

Trigonometric Functions Web

Series and Taylor Polynomials Web

Probability Web

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Find $\frac { d p } { d q }$ if $p = \ln \left( \frac { 7 q ^ { 2 } - 3 } { q } \right)$

Use a graphing utility to graph the function $f ( x ) = 3 ^ { - x ^ { 2 } }$ .

Use the properties of logarithms to expand $\ln \frac { 3 } { 10 }$

Find the derivative of $f ( x ) = x ^ { - 3 } - 10 e ^ { x }$

Find the relative minima, and use a graphing utility to check your results. $y = 3 \ln x - 5 x$

Determine whether the function below has any horizontal asymptotes. $f ( x ) = \frac { e ^ { x } - e ^ { - x } } { 2 }$

Locate any relative extrema and inflection points of the function $y = x ^ { 6 } \ln \frac { x } { 6 }$ .

Find an equation of the tangent line to the graph of $y = \log _ { 3 } x$ at the point $( 27,3 )$ .

Find the derivative of the following function. $y = \ln \left( 4 x ^ { 3 } - 9 x \right) - 3 x$

Use implicit differentiation to find $\frac { d y } { d x }$ . $9 e ^ { xy } + y ^ { 2 } = 15$

Find the equation of the tangent line to $f ( x ) = 9 x + e ^ { x }$ at the point (0,1).

Find $f ^ { \prime } ( t )$ if $f ( t ) = t ^ { 10 } 10 ^ { 3 t }$ .

Evaluate the expression $256 ^ { \frac { 2 } { 4 } }$ .

Sketch the graph of the function $f ( x ) = e ^ { 3 x }$ .

Find $y ^ { \prime }$ . $y = \ln ( 5 x + 7 ) ^ { 1 / 8 }$