Deck 4: Derivatives of Transcendental Functions

Question 1

Which of the following is identical to $\sin ^{2} 8 x+\cos ^{2} 8 x$ ?

A) 16
B) 4
C) 1
D) 8

Answer 1

4

Question 2

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

A) $\cos ^{2} x$
B) 1
C) $\tan ^{2} x$
D) $\sin ^{2} x$

Answer 2

$\tan ^{2} x$

Question 3

Which of the following is identical to $\sin (x+\pi)$ ?

A) $\cos x$
B) - $\cos \mathrm{x}$
C) $-\sin x$
D) $\sin \mathrm{x}$

Answer 3

$-\sin x$

Question 4

Which of the following is identical to $\cos (x+y) \cos (x-y)$ ?

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

Question 5

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

A) $\sin ^{2} x$
B) $\sin \mathrm{x} \cos \mathrm{x}$
C) $\tan ^{2} x$
D) $\cos ^{2} x$

Question 6

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

A) $x=\frac{1}{2} \cos ^{-1} 5 y$
B) $x=5 \cos ^{-1} 2 y$
C) $x=\frac{1}{5} \cos ^{-1} 2 y$
D) $x=\cos ^{-1} \frac{2 y}{5}$

Question 7

Find a value of $\arccos \left(\frac{-\sqrt{3}}{2}\right)$ in radians.

A) $\frac{5 \pi}{6}$
B) $\frac{\pi}{6}$
C) $\frac{\pi}{3}$
D) $\frac{2 \pi}{3}$

Question 8

Find the value of $\tan \left[\arcsin \left(\frac{1}{\sqrt{2}}\right)\right]$ .

A) $\frac{\pi}{4}$
B) 1
C) $\frac{1}{\sqrt{2}}$
D) $\sqrt{2}$

Question 9

Find the value of $\sec \left[\arccos \left(\frac{-1}{2}\right)\right]$ .

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

Question 10

Find an algebraic expression for $\tan (\arccos 3 x)$ .

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

Question 11

Solve for $x: \log 8 x=\frac{5}{3}$

A) 40.3
B) 32
C) 13.3
D) 3.48

Question 12

Solve for $x$ : $\log _{2} x=-1$

A) 0
B) $\frac{1}{2}$
C) 2
D) 1

Question 13

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

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

Question 14

Find the value of $8 \log 2(1 / 3)$ .

A) $\frac{1}{27}$
B) $\frac{1}{81}$
C) $\frac{1}{64}$
D) $\frac{1}{9}$

Question 15

Write $\log _{2} \frac{\sqrt{\mathrm{x}(\mathrm{x}+6)}}{\mathrm{x}^{4}}$ as a sum or difference of multiples of single logarithms.

A) $\frac{3 \log _{2} x}{2}+\log _{2}(x+6)$
B) $\frac{\log _{2} 6 x}{2}-3 \log _{2} x$
C) $\frac{\log _{2} x}{2}+\frac{\log _{2}(x+6)}{2}-4 \log _{2} x$
D) $\log _{2}(x+6)-3 \log _{2} x$

Question 16

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

A) $f^{\prime}(x)=3 \sin ^{2} x$
B) $f^{\prime}(x)=3 \sin ^{2} x \cos x$
C) $f^{\prime}(x)=-3 \sin ^{2} x \cos x$
D) $f^{\prime}(x)=2 \sin ^{2} x$

Question 17

Find the derivative of $f(x)=\sin 5 x \cos 4 x$ .

A) $f^{\prime}(x)=-5 \cos 5 x \cos 4 x+4 \sin 5 x \sin 4 x$
B) $f^{\prime}(x)=20 \cos 5 x \sin 4 x$
C) $f^{\prime}(x)=-20 \cos 5 x \sin 4 x$
D) $f^{\prime}(x)=5 \cos 5 x \cos 4 x-4 \sin 5 x \sin 4 x$

Question 18

Find the derivative of $f(x)=\sec ^{3} 2 x$ .

A) $f^{\prime}(x)=3 \sec ^{3} 2 x$
B) $f^{\prime}(x)=6 \sec ^{2} 2 x$
C) $f^{\prime}(x)=6 \sec ^{3} 2 x \tan 2 x$
D) $f^{\prime}(x)=3 \sec ^{3} 2 x \tan 2 x$

Question 19

Find the derivative of $f(x)=\frac{x}{\cos 2 x}$ .

A) $f^{\prime}(x)=\frac{\cos 2 x+x \sin 2 x}{\cos ^{2} 2 x}=\sec 2 x+x \sec 2 x \tan 2 x$
B) $f^{\prime}(x)=\frac{\cos 2 x-x \sin 2 x}{\cos ^{2} 2 x}=\sec 2 x-x \sec 2 x \tan 2 x$
C) $f^{\prime}(x)=\frac{\cos 2 x-2 x \sin 2 x}{\cos ^{2} 2 x}=\sec 2 x-2 x \sec 2 x \tan 2 x$
D) $f^{\prime}(x)=\frac{\cos 2 x+2 x \sin 2 x}{\cos ^{2} 2 x}=\sec 2 x+2 x \sec 2 x \tan 2 x$

Question 20

Find the derivative of $\mathrm{f}(\mathrm{x})=\frac{\sin \mathrm{x}}{1+\cos \mathrm{x}}$ .

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

Question 21

Find the derivative of $f(x)=\arctan (3 x)$ .

A) $f^{\prime}(x)=\frac{3}{1+3 x^{2}}$
B) $f^{\prime}(x)=\frac{3}{1+9 x^{2}}$
C) $f^{\prime}(x)=\frac{1}{1+3 x^{2}}$
D) $f^{\prime}(x)=\frac{1}{1+9 x^{2}}$

Question 22

Find the derivative of $f(x)=x \arcsin x$ .

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

Question 23

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

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

Question 24

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

A) $f^{\prime}(x)=e^{\sin x} \cdot(\cos x)$
B) $f^{\prime}(x)=e^{\cos x} \cdot(\cos x)$
C) $f^{\prime}(x)=e^{\cos x}$
D) $f^{\prime}(x)=e^{\sin x}$

Question 25

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

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

Question 26

Find the equation of the line tangent to the curve $f(x)=\ln x^{2}$ at the point $(1,0)$ .

A) $y=2 x-2$
B) $y=x-1$
C) $y=\frac{1}{x}(x-1)$
D) $y=\frac{2}{x}(x-1)$

Question 27

The current in a circuit is given by $i=4 t^{2} e^{8 / t}$ . Find the value of $t$ when $\frac{d i}{d t}=0$ .

A) $t=12$
B) $t=8$
C) $t=4$
D) $t=16$

Question 28

For the problems below, find the derivative of each function.

- $y=\cos 15 x$

Question 29

For the problems below, find the derivative of each function.

- $y=3 \sin (2 x+1)$

Question 30

For the problems below, find the derivative of each function.

- $y=\frac{3 x^{2}+5}{\sin 2 x}$

Question 31

For the problems below, find the derivative of each function.

- $y=\tan \sqrt{x^{2}+4 x}$

Question 32

For the problems below, find the derivative of each function.

- $y=\frac{\sec x}{x^{2}}$

Question 33

For the problems below, find the derivative of each function.

- $y=\sin 3 x \cot ^{2} x$

Question 34

For the problems below, find the derivative of each function.

- $y=(1+\csc x)^{3}$

Question 35

For the problems below, find the derivative of each function.

- $y=5 \arcsin (x-1)$

Question 36

For the problems below, find the derivative of each function.

- $y=6 \operatorname{arccot}\left(\frac{x}{2}\right)$

Question 37

For the problems below, find the derivative of each function.

- $y=4 \arccos \left(1-x^{2}\right)$

Question 38

For the problems below, find the derivative of each function.

- $y=\operatorname{arccsc}^{4} 2 x$

Question 39

For the problems below, find the derivative of each function.

- $y=\log (8 x+2)$

Question 40

For the problems below, find the derivative of each function.

- $y=\ln \left(5 x^{2}-15\right)$

Question 41

For the problems below, find $\frac{d y}{d x}$ using logarithmic differentiation.

- $y=\frac{(x-3)^{2}(5 x+4)}{(2 x-7)(6-x)}$

Question 42

For the problems below, find $\frac{d y}{d x}$ using logarithmic differentiation.

- $y=(\cos x)^{x}$

Question 43

For the problems below, find the derivative of each function.

- $y=e^{x^{3}+3 x}$

Question 44

For the problems below, find the derivative of each function.

- $y=\sin e^{2 x}$

Question 45

For the problems below, find the derivative of each function.

- $y=x^{2} e^{x}$

Question 46

For the problems below, find the derivative of each function.
-Find any relative maximums or minimums of the function $y=x \ln \left(x^{2}\right)$ .

Question 47

Find the indicated limits if they exist.

- $\lim _{x \rightarrow 1} \frac{x^{4}-1}{\ln x}$

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

Question 48

Find the indicated limits if they exist.

- $\lim _{t \rightarrow 0} t^{3} e^{4 t}$

A) 125
B) 4
C) $\frac{4}{25}$
D) 0

Question 49

Find the indicated limits if they exist.

- $\lim 8 \mathrm{x}^{-3 \mathrm{e}^{2}}$ $x \rightarrow 0$

A) $\frac{32}{3}$
B) $\frac{16}{3}$
C) 3
D) Does not exist.

Question 50

Write the word or phrase that best completes each statement or answers the question.

- $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cos x}{4 x-2 \pi}$

Question 51

Write the word or phrase that best completes each statement or answers the question.

- $\lim _{\mathrm{x} \rightarrow 0} \frac{\mathrm{e}^{\mathrm{x}}+\mathrm{x}-1}{4 \mathrm{x}^{2}}$

Question 52

Write the word or phrase that best completes each statement or answers the question.

- $\lim _{x \rightarrow 0} \frac{\sqrt[3]{\left(1+x^{3}\right)}}{x^{3}}$