Deck 2: The Derivative

Question 1

For $s=3 t^{2}-t+5$ , find $\bar{v}$ at $t=4, \Delta t=3$ .

A) 49
B) 23
C) 32
D) $48 . \overline{3}$

Answer 1

32

Question 2

Find $\lim _{x \rightarrow 2}\left(x^{2}-9\right)\left(x^{2}+5 x\right)$ .

A) 5
B) - 70
C) -5
D) 70

Answer 2

- 70

Question 3

Find the equation of the line tangent to $y=-5 x^{2}+3 x$ at (2, - 14).

A) $y=17 x-14$
B) $y=17 x+20$
C) $y=10 x+34$
D) $y=-10 x-6$

Answer 3

$y=17 x+20$

Question 4

Find the derivative of $y=\frac{1}{3 x-4}$ .

A) $\frac{d y}{d x}=\frac{-1}{(3 x-4)^{3}}$
B) $\frac{d y}{d x}=3$
C) $\frac{d y}{d x}=\frac{-3}{(3 x-4)^{2}}$
D) $\frac{d y}{d x}=\frac{3}{3 x-4}$

Question 5

Find the derivative of $y=\frac{4}{3} x^{2}-9 x^{2}+x-15$ .

A) $\frac{d y}{d x}=\frac{-72}{x^{6}}$
B) $\frac{d y}{d x}=4 x^{3}-18 x^{2}+x$
C) $\frac{d y}{d x}=8 x-18$
D) $\frac{d y}{d x}=4 x^{2}-18 x+1$

Question 6

Find the derivative of $y=\frac{4 x-8}{8 x+15}$ .

A) $\frac{d y}{d x}=\frac{64 x-4}{(8 x+15)^{2}}$
B) $\frac{d y}{d x}=\frac{124}{64 x^{2}+255}$
C) $\frac{d y}{d x}=\frac{124}{(8 x+15)^{2}}$
D) $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2}$

Question 7

Find the derivative of $y=\sqrt{3 x^{2}-8 x+5}$ .

A) $\frac{d y}{d x}=\frac{3 x-4}{3 x^{2}-8 x+5}$
B) $\frac{d y}{d x}=\frac{6 x-8}{\sqrt{3 x^{2}-8 x+5}}$
C) $\frac{d y}{d x}=\frac{3 x-4}{\sqrt{3 x^{2}-8 x+5}}$
D) $\frac{d y}{d x}=(3 x-4) \sqrt{3 x^{2}-8 x+5}$

Question 8

Use implicit differentiation to find the derivative of $y^{4}=8 x y^{2}-6 x$ .

A) $\frac{d y}{d x}=\frac{8 x y-3}{2 y^{3}}$
B) $\frac{d y}{d x}=\frac{4 y^{2}-3}{2 y^{3}-8 x y}$
C) $\frac{d y}{d x}=\frac{2 y^{3}-3}{2-8 x y}$
D) $\frac{d y}{d x}=\frac{-3}{2 y^{3}-4 x^{2}}$

Question 9

Find the fourth derivative of $y=3 x^{7}+8 x^{6}-9 x^{4}+5 x^{3}-4$ .

A) $y^{i v}=7560 x^{2}+7560 x$
B) $y^{i v}=7560 x^{3}+5760 x^{2}-216 x$
C) $y^{i v}=2520 x^{3}+2880 x^{2}-216$
D) $y^{i v}=630 x^{4}+960 x^{3}-216 x+30$

Question 10

Evaluate $f(x)=\frac{\left(x^{2}-3 x\right) \sqrt{x+5}}{x-4}$ at $x=6$ .

A) 89.5
B) 9
C) 29.8
D) 2.49

Question 11

Find $\lim _{x \rightarrow} \frac{8 x-3}{x}$ .

A) 5
B) -3
C) 8
D) 0

Question 12

Find the slope of the line tangent to the curve $y=3 x^{2}+9 x-5$ at the point $(-2,-11)$ .

A) -12
B) -9
C) -3
D) - 2

Question 13

Find the slope of the line tangent to $y=4 x^{2}-5 x+6$ at the point (-2, 32).

A) 11
B) 8
C) - 21
D) - 11

Question 14

Find the derivative of $y=\frac{16}{x^{8}}$ .

A) $\frac{d y}{d x}=\frac{-32}{x^{7}}$
B) $\frac{d y}{d x}=\frac{2}{x^{7}}$
C) $\frac{d y}{d x}=\frac{2}{x^{8}}$
D) $\frac{d y}{d x}=\frac{-128}{x^{9}}$

Question 15

Find the derivative of $y=\left(x^{2}-4 x+9\right)\left(3 x^{2}+10\right)$ .

A) $\frac{d y}{d x}=3 x^{3}-24 x^{2}+54 x-4$
B) $\frac{d y}{d x}=12 x^{2}+24 x$
C) $\frac{d y}{d x}=6 x^{3}-12 x^{2}+20 x-40$
D) $\frac{d y}{d x}=12 x^{3}-36 x^{2}+74 x-40$

Question 16

Find the derivative of $y=\frac{1}{\left(4 x^{3}-6 x+8\right)^{2}}$ .

A) $\frac{d y}{d x}=\frac{12 x^{2}-6}{\left(4 x^{3}-6 x+8\right)^{3}}$
B) $\frac{d y}{d x}=\frac{-12\left(2 x^{2}-1\right)}{\left(4 x^{3}-6 x+8\right)^{4}}$
C) $\frac{d y}{d x}=\frac{12 x^{2}-6}{\left(4 x^{3}-6 x+8\right)^{4}}$
D) $\frac{d y}{d x}=\frac{-12\left(2 x^{2}-1\right)}{\left(4 x^{3}-6 x+8\right)^{3}}$

Question 17

Use implicit differentiation to find the derivative of $\frac{x+y^{2}}{2 x-y}=4 x^{3}$ .

A) $\frac{d y}{d x}=\frac{24 x^{3}-12 x^{2} y+2 y^{2}}{y^{2}-x-4 x y}$
B) $\frac{d y}{d x}=\frac{4 x^{2}(2 x-y)^{2}-y^{2}}{4 x y+y^{2}-x}$
C) $\frac{d y}{d x}=\frac{12 x^{2}(2 x-y)^{2}+2 y^{2}+y}{4 x y+x-y^{2}}$
D) $\frac{d y}{d x}=\frac{4 x^{2}(2 x-y)^{2}-2 y^{2}-y}{4 x y+x-y^{2}}$

Question 18

Find the second derivative of $y=y^{2}-3 x^{4}+4$ .

A) $y^{\prime \prime}=\frac{72 x^{2} y-36 x^{2}-288 x^{6}}{(1-2 y)^{3}}$
B) $y^{\prime \prime}=\frac{\left(72 x^{2} y-36 x^{2}\right)(1-2 y)-288 x^{6}}{(1-2 y)^{2}}$
C) $y^{\prime \prime}=\frac{(1-2 y)\left(36 x^{2} y-18 x^{2}\right)}{(1-2 y)^{3}}$
D) $y^{\prime \prime}=\frac{\left(72 x^{2} y-36 x^{2}\right)(1-2 y)-288 x^{6}}{(1-2 y)^{3}}$

Question 19

For the problems below, evaluate each function at the given value.

-19 $f(t)=2 t^{3}+t-6 ; t=4$

Question 20

For the problems below, evaluate each function at the given value.
-20 $f(x)=\sqrt{x^{2}-4 x} ; x=5$

Question 21

For the problems below, find: a) $\Delta \mathrm{s}$ and b) $\overline{\mathrm{v}}$ for each function expressing distance $\mathrm{s}$ in terms of time $\mathrm{t}$ . 21) $s=5 t+3$
-s = 5t + 3

Question 22

For the problems below, find: a) $\Delta \mathrm{s}$ and b) $\overline{\mathrm{v}}$ for each function expressing distance $\mathrm{s}$ in terms of time $\mathrm{t}$ . 21) $s=5 t+3$
-22 $s=3 t^{3}-10 t+3$

Question 23

For the problems below, find: a) $\Delta \mathrm{s}$ and b) $\overline{\mathrm{v}}$ for each function expressing distance $\mathrm{s}$ in terms of time $\mathrm{t}$ . 21) $s=5 t+3$
-Given $s=4 t^{2}+16 t-5$ , find $\bar{v}$ at $t=3$ and $\Delta t=1$ , where $s$ is in metres and $t$ is in seconds.

Question 24

For the problems below, find: a) $\Delta \mathrm{s}$ and b) $\overline{\mathrm{v}}$ for each function expressing distance $\mathrm{s}$ in terms of time $\mathrm{t}$ . 21) $s=5 t+3$
-Given $s=f(t)=4 t^{2}$ - 9, find the instantaneous velocity $v$ of an object moving along a straight line when $t=3$ . The distance, $s$ , is in metres and $t$ is in seconds.

Question 25

For the problems below, find each limit.

- $\lim _{x \rightarrow 5} \frac{x^{2}-25}{x-5}$

Question 26

For the problems below, find each limit.

- $\lim _{\mathrm{x} \rightarrow} \frac{4 \mathrm{x}^{2}-7}{8 \mathrm{x}^{2}+5}$

Question 27

For the problems below, find each limit.

- $\lim _{\mathrm{x} \rightarrow 8} 5$

Question 28

For the problems below, find each limit.

- $\lim _{\mathrm{x} \rightarrow / 2} \frac{2 \mathrm{x}^{2}-9 \mathrm{x}-18}{2 \mathrm{x}+3}$

Question 29

For the problems below, find each limit.

- $\lim _{\mathrm{x} \rightarrow \infty} \frac{1}{\mathrm{x}^{3}}$

Question 30

For the problems below, find each limit.

- $\lim _{x \rightarrow \infty}\left(3 x^{2}-2 x\right)$

Question 31

Find the slope of the line tangent to the curve $y=x^{2}-4 x+7$ at $(5,12)$ .

Question 32

Find the equation of the line tangent to the curve $y=3 x^{2}-9$ at (4, 39).

Question 33

Find the point or points on the curve $y=2 x^{3}-5 x^{2}$ where the slope of the tangent line is 0 .

Question 34

For the problems below, find the derivative of each function.
- $y=2 x^{2}-5 x$

Question 35

For the problems below, find the derivative of each function.
- $y=\frac{1}{2 x+7}$

Question 36

Find the equation of the line tangent to $y=x^{3}$ at the point (1, 8).

Question 37

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

- $y=x^{14}$

Question 38

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

- $y=2 x^{4}-3 x^{2}+7$

Question 39

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

- $y=\sqrt{6} x^{4}$

Question 40

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

- $y=\frac{2}{x^{8}}$

Question 41

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

- $y=\frac{1}{x^{2} \sqrt{\mathrm{x}}}$

Question 42

For $f(x)=3 x^{4}-x^{2}+5$ , findf'(2).

Question 43

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

- $y=\left(x^{4}-2 x^{3}\right)\left(x^{3}+5 x\right)$

Question 44

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

- $y=x^{8}\left(3 x^{2}+5 x-7\right)$

Question 45

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

- $y=\frac{x}{4 x-9}$

Question 46

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

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

Question 47

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

- $y=(3 x-7)^{9}$

Question 48

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

- $y=\frac{2}{\left(x^{3}-8\right)^{5}}$

Question 49

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

- $y=\frac{(4 x-1)^{6}}{(3 x+2)^{4}}$

Question 50

For the problems below, find the derivative using implicit differentiation.

- $8 x+3 y=6$

Question 51

For the problems below, find the derivative using implicit differentiation.

- $9 x^{2}+4 y^{2}=36$

Question 52

For the problems below, find the derivative using implicit differentiation.

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

Question 53

For the problems below, find the derivative using implicit differentiation.

- $y=3 x y^{2}+\sqrt{x-1}$

Question 54

Find the equation of the line tangent to the curve $3 x y-5 y^{2}=10$ at $(5,2)$ .

Question 55

Given $y=4 x^{5}-3 x^{4}+8 x^{2}$ , find the first three derivatives of $y$ .

Question 56

Find $y^{\prime}$ and $y^{\prime \prime}$ implicitly and express the result in $x$ and $y$ . $(2 \mathrm{y}+3)^{2}=8 \mathrm{x}$