Question 1

For the problems below, find the derivative of each function.
-  $y=5 \arcsin (x-1)$

Accepted Answer

$y^{\prime}=\frac{5}{\sqrt{2 x-x^{2}}}$

Question 2

Which of the following is identical to $\frac{1}{2}(1-\cos 2 x)$ ?

Accepted Answer

A)    sin⁡2x\sin ^{2} xsin2x 
B)    sin⁡xcos⁡x\sin \mathrm{x} \cos \mathrm{x}sinxcosx 
C)    tan⁡2x	an ^{2} xtan2x 
D)    cos⁡2x\cos ^{2} xcos2x A

Question 3

Find the derivative of $f(x)=x^{2} e^{4 x}$.

Accepted Answer

A)    f′(x) =4x2e4x+2xe4x(2x+1) f^{\prime}(x) =4 x^{2} e^{4 x}+2 x e^{4 x}(2 x+1) f′(x) =4x2e4x+2xe4x(2x+1)  
B)    f′(x) =8xe4xf^{\prime}(x) =8 x e^{4 x}f′(x) =8xe4x 
C)    f′(x) =2xe4xf^{\prime}(x) =2 x e^{4 x}f′(x) =2xe4x 
D)    f′(x) =x2e4x+2xe4x=xe4x(x+2) f^{\prime}(x) =x^{2} e^{4 x}+2 x e^{4 x}=x e^{4 x}(x+2) f′(x) =x2e4x+2xe4x=xe4x(x+2)  A

Question 4

For the problems below, find the derivative of each function.
- $y=\sin 3 x \cot ^{2} x$

Accepted Answer

The answer of For the problems below, find the derivative...

Question 5

Write the word or phrase that best completes each statement or answers the question.
- $\lim _{x \rightarrow 0} \frac{\sqrt[3]{\left(1+x^{3}\right)}}{x^{3}}$

Accepted Answer

The answer of Write the word or phrase that best...

Question 6

Solve for $x: y=\frac{1}{2}(\cos 5 x)$

Accepted Answer

A)    x=12cos⁡−15yx=\frac{1}{2} \cos ^{-1} 5 yx=21​cos−15y 
B)    x=5cos⁡−12yx=5 \cos ^{-1} 2 yx=5cos−12y 
C)    x=15cos⁡−12yx=\frac{1}{5} \cos ^{-1} 2 yx=51​cos−12y 
D)    x=cos⁡−12y5x=\cos ^{-1} \frac{2 y}{5}x=cos−152y​

Question 7

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

Accepted Answer

A)    f′(x) =3x2x3+5f^{\prime}(x) =\frac{3 x^{2}}{x^{3}+5}f′(x) =x3+53x2​ 
B)    f′(x) =13x2f^{\prime}(x) =\frac{1}{3 x^{2}}f′(x) =3x21​ 
C)    f′(x) =3x2ln⁡(x3+5) f^{\prime}(x) =\frac{3 x^{2}}{\ln \left(x^{3}+5\right) }f′(x) =ln(x3+5) 3x2​ 
D)    f′(x) =1x3+5f^{\prime}(x) =\frac{1}{x^{3}+5}f′(x) =x3+51​

Question 8

For the problems below, find the derivative of each function.
- $y=\operatorname{arccsc}^{4} 2 x$

Accepted Answer

The answer of For the problems below, find the derivative...

Question 9

For the problems below, find the derivative of each function.
- $y=3 \sin (2 x+1)$

Accepted Answer

The answer of For the problems below, find the derivative...

Question 10

For the problems below, find the derivative of each function.
-  $y=\sin e^{2 x}$

Accepted Answer

The answer of For the problems below, find the derivative...

Question 11

Find the value of $\log _{4} 64$.

Accepted Answer

A)   24
B)   4
C)   3
D)   9

Question 12

Which of the following is identical to $\sin ^{2} x+\sin ^{2} x \tan ^{2} x$ ?

Accepted Answer

A)    cos⁡2x\cos ^{2} xcos2x 
B)   1
C)    tan⁡2x	an ^{2} xtan2x 
D)    sin⁡2x\sin ^{2} xsin2x

Question 13

Find the indicated limits if they exist.
-  $\lim _{x \rightarrow 1} \frac{x^{4}-1}{\ln x}$

Accepted Answer

A)   0
B)   1
C)   24
D)   4

Question 14

For the problems below, find the derivative of each function.
- $y=\frac{\sec x}{x^{2}}$

Accepted Answer

The answer of For the problems below, find the derivative...

Question 15

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

Accepted Answer

A)    f′(x) =3sin⁡2xf^{\prime}(x) =3 \sin ^{2} xf′(x) =3sin2x 
B)    f′(x) =3sin⁡2xcos⁡xf^{\prime}(x) =3 \sin ^{2} x \cos xf′(x) =3sin2xcosx 
C)    f′(x) =−3sin⁡2xcos⁡xf^{\prime}(x) =-3 \sin ^{2} x \cos xf′(x) =−3sin2xcosx 
D)    f′(x) =2sin⁡2xf^{\prime}(x) =2 \sin ^{2} xf′(x) =2sin2x

Question 16

For the problems below, find the derivative of each function.
- $y=\tan \sqrt{x^{2}+4 x}$

Accepted Answer

The answer of For the problems below, find the derivative...

Question 17

For the problems below, find the derivative of each function.
-  $y=e^{x^{3}+3 x}$

Accepted Answer

The answer of For the problems below, find the derivative...

Question 18

For the problems below, find the derivative of each function.
-  $y=x^{2} e^{x}$

Accepted Answer

The answer of For the problems below, find the derivative...

Question 19

Write the word or phrase that best completes each statement or answers the question.
- $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cos x}{4 x-2 \pi}$

Accepted Answer

The answer of Write the word or phrase that best...

Question 20

Find the indicated limits if they exist.
-  $\lim _{t \rightarrow 0} t^{3} e^{4 t}$

Accepted Answer

A)   125
B)   4
C)    425\frac{4}{25}254​ 
D)   0

Exam 4: Derivatives of Transcendental Functions

For the problems below, find the derivative of each function. - $y=5 \arcsin (x-1)$

Which of the following is identical to $\frac{1}{2}(1-\cos 2 x)$ ?

Find the derivative of $f(x) =x^{2} e^{4 x}$ .

For the problems below, find the derivative of each function. - $y=\sin 3 x \cot ^{2} x$

Write the word or phrase that best completes each statement or answers the question. - $\lim _{x \rightarrow 0} \frac{\sqrt[3]{\left(1+x^{3}\right)}}{x^{3}}$

Solve for $x: y=\frac{1}{2}(\cos 5 x)$

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

For the problems below, find the derivative of each function. - $y=\operatorname{arccsc}^{4} 2 x$

For the problems below, find the derivative of each function. - $y=3 \sin (2 x+1)$

For the problems below, find the derivative of each function. - $y=\sin e^{2 x}$

Find the value of $\log _{4} 64$ .

Which of the following is identical to $\sin ^{2} x+\sin ^{2} x \tan ^{2} x$ ?

Find the indicated limits if they exist. - $\lim _{x \rightarrow 1} \frac{x^{4}-1}{\ln x}$

For the problems below, find the derivative of each function. - $y=\frac{\sec x}{x^{2}}$

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

For the problems below, find the derivative of each function. - $y=\tan \sqrt{x^{2}+4 x}$

For the problems below, find the derivative of each function. - $y=e^{x^{3}+3 x}$

For the problems below, find the derivative of each function. - $y=x^{2} e^{x}$

Write the word or phrase that best completes each statement or answers the question. - $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cos x}{4 x-2 \pi}$

Find the indicated limits if they exist. - $\lim _{t \rightarrow 0} t^{3} e^{4 t}$

Exam 4: Derivatives of Transcendental Functions

For the problems below, find the derivative of each function. - y=5arcsin⁡(x−1)y=5 \arcsin (x-1)y=5arcsin(x−1)

Which of the following is identical to 12(1−cos⁡2x) \frac{1}{2}(1-\cos 2 x) 21​(1−cos2x) ?

Find the derivative of f(x) =x2e4xf(x) =x^{2} e^{4 x}f(x) =x2e4x .

For the problems below, find the derivative of each function. - y=sin⁡3xcot⁡2xy=\sin 3 x \cot ^{2} xy=sin3xcot2x

Write the word or phrase that best completes each statement or answers the question. - lim⁡x→0(1+x3)3x3\lim _{x \rightarrow 0} \frac{\sqrt[3]{\left(1+x^{3}\right)}}{x^{3}}limx→0​x33(1+x3)​​

Solve for x:y=12(cos⁡5x) x: y=\frac{1}{2}(\cos 5 x) x:y=21​(cos5x)

Find the derivative of f(x) =ln⁡(x3+5) f(x) =\ln \left(x^{3}+5\right) f(x) =ln(x3+5) .

For the problems below, find the derivative of each function. - y=arccsc⁡42xy=\operatorname{arccsc}^{4} 2 xy=arccsc42x

For the problems below, find the derivative of each function. - y=3sin⁡(2x+1)y=3 \sin (2 x+1)y=3sin(2x+1)

For the problems below, find the derivative of each function. - y=sin⁡e2xy=\sin e^{2 x}y=sine2x

Find the value of log⁡464\log _{4} 64log4​64 .

Which of the following is identical to sin⁡2x+sin⁡2xtan⁡2x\sin ^{2} x+\sin ^{2} x \tan ^{2} xsin2x+sin2xtan2x ?

Find the indicated limits if they exist. - lim⁡x→1x4−1ln⁡x\lim _{x \rightarrow 1} \frac{x^{4}-1}{\ln x}limx→1​lnxx4−1​

For the problems below, find the derivative of each function. - y=sec⁡xx2y=\frac{\sec x}{x^{2}}y=x2secx​

Find the derivative of f(x) =sin⁡3xf(x) =\sin ^{3} xf(x) =sin3x .

For the problems below, find the derivative of each function. - y=tan⁡x2+4xy=\tan \sqrt{x^{2}+4 x}y=tanx2+4x​

For the problems below, find the derivative of each function. - y=ex3+3xy=e^{x^{3}+3 x}y=ex3+3x

For the problems below, find the derivative of each function. - y=x2exy=x^{2} e^{x}y=x2ex

Write the word or phrase that best completes each statement or answers the question. - lim⁡x→π2cos⁡x4x−2π\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cos x}{4 x-2 \pi}limx→2π​​4x−2πcosx​

Find the indicated limits if they exist. - lim⁡t→0t3e4t\lim _{t \rightarrow 0} t^{3} e^{4 t}limt→0​t3e4t

For the problems below, find the derivative of each function. - $y=5 \arcsin (x-1)$

Which of the following is identical to $\frac{1}{2}(1-\cos 2 x)$ ?

Find the derivative of $f(x) =x^{2} e^{4 x}$ .

For the problems below, find the derivative of each function. - $y=\sin 3 x \cot ^{2} x$

Write the word or phrase that best completes each statement or answers the question. - $\lim _{x \rightarrow 0} \frac{\sqrt[3]{\left(1+x^{3}\right)}}{x^{3}}$

Solve for $x: y=\frac{1}{2}(\cos 5 x)$

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

For the problems below, find the derivative of each function. - $y=\operatorname{arccsc}^{4} 2 x$

For the problems below, find the derivative of each function. - $y=3 \sin (2 x+1)$

For the problems below, find the derivative of each function. - $y=\sin e^{2 x}$

Find the value of $\log _{4} 64$ .

Which of the following is identical to $\sin ^{2} x+\sin ^{2} x \tan ^{2} x$ ?

Find the indicated limits if they exist. - $\lim _{x \rightarrow 1} \frac{x^{4}-1}{\ln x}$

For the problems below, find the derivative of each function. - $y=\frac{\sec x}{x^{2}}$

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

For the problems below, find the derivative of each function. - $y=\tan \sqrt{x^{2}+4 x}$

For the problems below, find the derivative of each function. - $y=e^{x^{3}+3 x}$

For the problems below, find the derivative of each function. - $y=x^{2} e^{x}$

Write the word or phrase that best completes each statement or answers the question. - $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cos x}{4 x-2 \pi}$

Find the indicated limits if they exist. - $\lim _{t \rightarrow 0} t^{3} e^{4 t}$