Question

Arithmetic Derivative (cont)

Answer 1

from http://openstudy.com/study#/updates/5471cd80e4b001ceb26b43cf

Answer 2

\[
n = \prod_{i=1}^kp_i^{a_i}
\]

Answer 3

\[
n'= \sum_{i=1}^ka_i\frac{n}{p_i}
\]

Answer 4

\[
n = a^xb^yc^z
\]\[
n' = xa^{x-1}b^yc^z+ya^{x}b^{y-1}c^z+za^{x}b^yc^{z-1} \\
=\left(x\frac{n}{a} + y\frac{n}{b} + z\frac{n}{c}\right)(a^{x-1}b^{y-1}c^{z-1})
\]

Answer 5

\[
u=\left(xbc+ yaz + zab\right)
\]

Answer 6

\[
n = a^xb^yz^zd^0
\]

Answer 7

if we have something like the max= 10 then we have

```
primes = [2, 3, 5, 7];
exps = [0, 0, 0, 0];

```
so we could count like this:

1 = [0,0,0,0]
2 = [1,0,0,0]
4 = [2,0,0,0]
8 = [3,0,0,0]
3 = [0,1,0,0]
6 = [1,1,0,0]
9 = [0,2,0,0]
5 = [0,0,1,0]
10 = [1,0,1,0]
7 = [0,0,0,1]

Answer 8

http://pastebin.com/E4y685z4

Answer 9

Hey, is your skype working?

Answer 10

\[
f'(u)
\]

Answer 11

\[
\int f(u)\;du
\]

Answer 12

\[
\int_0^xf(u)\;du
\]

Answer 13

\[
f(x) = x^2

\]

Answer 14

\[
u= x^2
\]

Answer 15

\[
f(u) = u^2
\]

Answer 16

\[
f(x) = x^2 = u
\]

Answer 17

\[
f'(2)
\]

Answer 18

\[
f'(x) = 2x \implies f'(2) = 2(2) = 4
\]

Answer 19

\[
\frac{d}{dx}f(2)
\]

Answer 20

\[
f'(2) = 4
\]

Answer 21

\[
\frac{d}{dx}f(2) = \frac{d}{dx}2^2 = 0
\]

Answer 22

\[f(t)=f(x,y)\]

Answer 23

\[f(x(t),y(t))\]

Answer 24

\[
r(t) = (x(t),y(t))
\]

Answer 25

\[
f\circ r
\]

Answer 26

\[
f(r(t))
\]

Answer 27

\(f(r)\)

Answer 28

\[
dx
\]

Answer 29

\[
\frac{d}{dx} = \frac{d}{du}\frac{du}{dx}
\]

Answer 30

\[
du = \frac{du}{dx}dx
\]

Answer 31

\[
f(x,y)
\]\[
y = g(x)
\]

Answer 32

\[
f(x,g(x))
\]

Answer 33

\[
\frac{d}{dx} f(x,y) = \frac{\partial }{\partial x}f(x,y) + \frac{\partial }{\partial y}f(x,y)\frac{d}{dx}y
\]

Answer 34

\[
\frac{d}{d(x^2)}f(x)
\]

Answer 35

\[
\frac{d}{dx}f(\sqrt x)
\]

Answer 36

\[
\frac{d}{d(x^2+1)}f(x)
\]

Answer 37

\[
\frac{d}{d(x^2+1)}
\]

Answer 38

\[
\frac{d}{d(x^2+1)} = \frac{d}{(2x)dx}
\]

Answer 39

\[
\frac{1}{2x}\frac{d}{dx}f(x)
\]

Answer 40

\[
\frac{d}{d(g(x))}f(x) = \frac{1}{g'(x)}\frac{d}{dx}f(x)
\]

Answer 41

\[
\frac{d}{dg}f(x)=\frac{\frac{d}{dx}f(x)}{\frac{d}{dx}g(x)}
\]

Answer 42

\[
\frac{df}{dg} \frac{dg}{dx}= \frac{df}{dx}
\]

Answer 43

\[
\frac{d}{d(5)}f(x)
\]

Answer 44

\[
\frac{\ldots}{5-5}
\]

Answer 45

\[
\lim_{x\to a}f(x)
\]

Answer 46

\[
\lim_{x-a\to 0}f(x-a)
\]

Answer 47

\[
\lim ^0 [f(x-a)]
\]

Answer 48

\(\color{blue}{\text{Originally Posted by}}\) @wio
\[
\lim ^0^+ [f(x-a)]
\]
\(\color{blue}{\text{End of Quote}}\)

Answer 49

\[
|x|<\delta \implies |f(x-a)-L|<\epsilon
\]

Answer 50

\[
g(x) = f(x-a)\\
|x|<\delta \implies |g(x)-L|<\epsilon
\]

Answer 51

\[
\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}
\]

Answer 52

\[\lim_{h \to 0} \int\limits_h^{x+h}\frac{f(x)}{h}dx = f'(x)\]

Answer 53

\[
\frac{dg}{dx}
\]

Answer 54

\[
d_xg
\]

Answer 55

\[\lim_{h \to 0} \int\limits_{f(x)}^{f(x+h)}\frac{dx}{h}=f'(x)\]

Answer 56

\[
\frac{df}{dx} \neq \frac{1}{\frac{dx}{df}}
\]

Answer 57

\[
f(x) = y\\
x = f^{-1}(y)\\
1 = f^{-1}\ '(y) y'(x)\\
1=f^{-1}\ '(f(x))y'(x) \\
\frac{1}{f^{-1}\ '(f(x))} = y'(x)
\]

Answer 58

\[
L_0
\]

Answer 59

f(x,y)

Answer 60

\[XYf(x,y)\]

Answer 61

\[
\lim_{(x,y)\to (0,0)}f(x,y) \neq \lim_{x\to 0}\left(\lim_{y\to 0}f(x,y)\right)
\]

Answer 62

\[
L
\]

Answer 63

\[
K\ \lim_{\to 0}\\
S\lim_{\to \infty}
\]

Answer 64

\[
\lim_{x\to 0^+}\frac 1x =+\infty
\]

Answer 65

\(x-a\)

Answer 66

\[
|x-a| <\delta
\]

Answer 67

\[
|x| = \sqrt{x^2}
\]

Answer 68

\[
|z| = \sqrt{z\bar z }
\]

Answer 69

\[
f(z)
\]

Answer 70

\[
d(x,a)
\]

Answer 71

\[
d(x,a) <\delta \implies d(f(x),L) <\epsilon
\]

Answer 72

\[
d(x,a)=|x-a|
\]

Answer 73

\[
d((x,y),(a.b)) = |x-a|+|x-b|
\]

Answer 74

\(\color{blue}{\text{Originally Posted by}}\) @wio
\[
d((x,y),(a.b)) = |x-a|+|y-b|
\]
\(\color{blue}{\text{End of Quote}}\)

Answer 75

\[
=\sqrt{(x-a)^2+(y-b)^2}
\]

Answer 76

\[
\sqrt[n]{(x-a)^n+(y-b)^n}
\]

Answer 77

\[
|z| = \sqrt{z\bar z}
\]

Answer 78

\[
\sqrt{a^2+b^2}
\]

Answer 79

\[
f(x,y)
\]

Answer 80

\[
\omega = -1
\]

Answer 81

\[
-7 = 7\omega
\]

Answer 82

\[
i^2 = \omega
\]

Answer 83

\[
\sqrt[n]{x}
\]

Answer 84

\[
z^4 =1
\]

Answer 85

\[
z\in\{0,i,-1,-i\}
\]

Answer 86

\[\LARGE 16=2^4e^{i 2 \pi }\] \[\LARGE 16^{1/4}=2e^{i \pi/2 } = 2i\]

Answer 87

\[\LARGE (e^{ \pi /2})^1, (e^{ \pi /2})^2, (e^{ \pi /2})^3, (e^{ \pi /2})^4\]

Answer 88

\[\LARGE (e^{ \pi /2})^0, (e^{ \pi /2})^1, (e^{ \pi /2})^2, (e^{ \pi /2})^3\]

Answer 89

\[
\omega^4 _1
\]\[
\omega_4^4
\]

Answer 90

\[
w_4^1 = w_4
\]

Answer 91

\[
z^n =1
\] \[
z=\{i\in\mathbb N, i \leq n|\omega _n\}
\]

Answer 92

\[\Large \omega_n = e^{ i \frac{2\pi}{n}}\]

Answer 93

\[
z^n = \alpha
\]