Deck 27: Derivatives of Algebraic Functions

Question 1

Evaluate the limit: $\lim_{x \rightarrow 0}(2x^2 - 9x + 5)$

Answer 1

5

Question 2

Evaluate the limit:
$\lim _{x \rightarrow 0}\left[\frac{x}{x^{2}-2}+\frac{1}{x^{2}}\right]$

Answer 2

$\infty$

Question 3

Evaluate the limit: $\lim_{x\rightarrow 0} xe^x$

Answer 3

0

Question 4

Evaluate the limit: $\lim_{x\rightarrow 0} \frac{x^2-\sqrt{x+d}}{d^2-x(x+d)}$

Question 5

Evaluate the limit: $\lim _{x \rightarrow 3^+} \frac{x^{2}+x-1}{x^{2}-2 x-3}$

Question 6

Evaluate the limit:
$\lim _{\pi/2}{\frac{\cos x}{\cot x}}$

Question 7

Evaluate the limit: $\lim _{x \rightarrow-2} \frac{x+2}{x^{2}+6 x+8}$

Question 8

Evaluate the limit:
$\lim_{x\rightarrow 2}{\frac{x^{4}-16}{x-2}}$

Question 9

Evaluate the limit: $$
$\lim_{x \rightarrow \infty} \frac{2 x^{2}+x-2}{x^{3}-17}$

Question 10

Evaluate the limit: $\lim _{x \rightarrow \infty} \frac{4 x^{2}+5 x+7}{2-3 x-8 x^{2}}$

Question 11

Evaluate the limit: $\lim _{x \rightarrow 3} \frac{9-x^{2}}{x-3}$

Question 12

Evaluate the limit:
$\lim _{x \rightarrow 2^{+}} \frac{2 x+3}{x-2}$

Question 13

Evaluate the limit: $\lim _{x \rightarrow 1.5} \frac{2 x^{2}+7 x-15}{4 x^{2}-9}$

Question 14

Evaluate the limit: $\lim_{x \rightarrow-1^{-}} \frac{x^2}{x+1}$

Question 15

The displacement of a mass on a spring is given by $s=\sin \frac{\pi}{2} t$ where $t$ is in seconds. Find the average velocity of the mass between
a) $t=0 \mathrm{~s}$ and $t=0.5 \mathrm{~s}$
b) $t=0.5 \mathrm{~s}$ and $t=1.0 \mathrm{~s}$

Question 16

Find the derivative by the delta method: $y=\frac{1}{1-x}$

Question 17

Find the derivative by the delta method: $\sqrt{2 x+3}$

Question 18

Using the delta method, find the instantaneous rate of change of the function at the given value of $x$ :
$y=\frac{x^{2}+1}{x} at x=2$

Question 19

Using the delta method, find the instantaneous rate of change of the function: $y=2 x^{3}+5 x$ , when $x=0.5$

Question 20

Find the derivative by the delta method: $y=\frac{x}{x+1}$

Question 21

Find the derivative by the delta method: $y=3 x^{2}-x^{3}$

Question 22

Find the derivative by the delta method: $y=x^{2}-2 x+1$

Question 23

Find the derivative by the delta method: $y=\frac{1}{x^{2}}$

Question 24

Find $\frac{d y}{d x}: y=\frac{x}{\sqrt{x}}+x^{2}$

Question 25

Solve: $\frac{d}{d x}\left(4 x^{4}+2 x^{3}-x+1\right)$

Question 26

Find $y^{\prime}$ if $y=\frac{3 x^{3}-\sqrt{x}}{x}$ .

Question 27

Find $y^{\prime}$ if $y=5 \sqrt[4]{x^{3}}$ .

Question 28

If $y=2 x^{3}-5 x^{2}$ , find $y^{\prime}(1 / 2)$ .

Question 29

Find the derivative: $y=(7-3 x)(x+5)$

Question 30

Find the derivative: $y=x-\sqrt[3]{x^{2}}$

Question 31

If $f(x)=x^{3 / 2}+x^{2 / 3}$ , find $f^{\prime}(1)$ .

Question 32

Find the derivative:
$y=4 x^{3}+8 \sqrt{x}-\frac{2}{x^{2}}+25$

Question 33

Find the derivative: $y=\frac{x^{3}}{5}-\frac{4}{x}$

Question 34

Given $f(x)=4 x^{2}-\frac{5}{x}$ , find $f^{\prime}(2)$ .

Question 35

Find the derivative: $y=(x-2)^{2}$

Question 36

Find the derivative: $y=\frac{3}{\sqrt[4]{x}}$

Question 37

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

Question 38

Find the derivative: $y=\sqrt[3]{\left(x^{3}+2 x\right)^{2}}$

Question 39

Find the derivative: $y=\sqrt{5-9 x^{2}}$

Question 40

If $y=\frac{1}{\left(x^{2}-2\right)^{3}}$ , find $y^{\prime}(-3$ . $) .$

Question 41

Find the derivative: $y=\sqrt{1-\frac{2}{x}}$

Question 42

Find the derivative: $y=\sqrt[3]{\left(\frac{3}{x}-x^{3}\right)^{2}}$

Question 43

Find the derivative: $y=5 x^{2}-6 \sqrt[4]{3-2 x}$

Question 44

Find the value of the derivative of $y=2\left(x^{2}-5\right)^{6}$ at $x=2.5$ .

Question 45

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

Question 46

Find the derivative: $y=4 \sqrt{5 x-3}$

Question 47

Find the value of the derivative of $y=\left(\frac{x^{2}}{4}-\frac{2}{x}\right)^{3}$ at $x=3$ . Use two decimal places.

Question 48

Find the derivative of: $y=\frac{3}{2(4 x-1)}-5 x^{2}$

Question 49

Find the derivative: $y=(2 x-1) \sqrt{3-4 x}$

Question 50

Find the derivative: $y=(x+4)^{2} \sqrt{x-3}$

Question 51

Find the slope of the tangent to the curve $y=4 x\left(x^{2}-2\right)^{2}$ at $x=1$ .

Question 52

Find the derivative: $y=\frac{x^{2}-1}{x^{2}+1}$

Question 53

Find the slope of the tangent to $y=\frac{x+1}{x-1}$ at $x=4$ .

Question 54

Find the derivative: $y=\sqrt{\frac{x^{2}+1}{x^{2}-1}}$

Question 55

If $f(x)=\frac{2 x-1}{\sqrt{x+1}}$ , find $f^{\prime}(3)$ .

Question 56

If $f(x)=4 x(3 x-2)^{3} \sqrt{x+2}$ , find $f^{\prime}(0.5)$ . Use three significant digits.

Question 57

Find the value of the derivative of $y=\frac{3 x+2}{\sqrt{x+1}}$ at $x=2$ . Give an exact answer.

Question 58

Find the derivative of: $y=7 x^{3}(2 x+1)^{4}$

Question 59

Find the derivative of: $y=\frac{8 x^{2}}{1-4 x}$

Question 60

Find the value of the derivative of $y=\frac{3(2 x+1)^{2}}{4(1-2 x)^{3}}$ at $x=2$ .

Question 61

Find the derivative:
$y=2 x^{2}-\frac{5 x-1}{3-2 x}$

Question 62

Find the derivative: $y=2(4 x+1)^{5}(3-2 x)^{6}$

Question 63

Find the derivative: $y=9 x^{4}(8-x)^{7}$

Question 64

If $y=u^{3}+4$ , find $\frac{d y}{d x}$ .

Question 65

Find $\frac{d y}{d x}$ if $y^{2}-2 x y+x=5$ .

Question 66

Find $\frac{d y}{d x}$ if $\left(x^{3}-y^{3}\right)^{2}=6 y$ .

Question 67

If $z=\left(w^{2}+2 w\right)^{3}$ , find $\frac{d z}{d x}$ .

Question 68

If $w=z+y^{3}+y+x$ , find $\frac{d w}{d x}$ .

Question 69

Find the derivative of $\frac{d}{d x}\left(3 x^{2}-4 x y^{4}-y^{3}\right)$ .

Question 70

Find $\frac{d y}{d x}$ if $x^{3}+y^{3}=x y$ .

Question 71

Find the value of the derivative of $x^{2}+y^{2}=25$ at the point $(-3,4)$ .

Question 72

Find $\frac{d y}{d x}$ if $2 x^{3}-x^{2} y=4 y^{2}-5$ .

Question 73

Find the value of the derivative of $2(x-y)^{3}=x y$ at $(2,1)$ .

Question 74

Find $\frac{d y}{d x}$ if $x^{3 / 2}+y^{3 / 2}=100$ .

Question 75

Find $\frac{d y}{d x}$ if $\sqrt{x y-1}=y^{2}-x^{2}$

Question 76

Find the second derivative of $y=\frac{x}{\sqrt{x-1}}$ .

Question 77

Find the third derivative of $y=(x+3)^{3}(x-2)^{2}$ .

Question 78

Find the second derivative of $y=\frac{x}{x^{2}-1}$ .

Question 79

Find the second derivative: $y=2\left(x^{2}+1\right)^{4}$

Question 80

Given $f(x)=x^{2}-x^{3}$ , find $f^{\prime \prime}(0.25)$ .