## anonymous 2 years ago why does: y+dy=(x+dx)^-2 equal: =x^-2(1+dx/x)^-2

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1. anonymous

put x in evidence

2. atlas

Another way of thinking is divide the expression on R.H.S with x^-2 and then multiply it with x^-2 on the outside

3. atlas

multiplying and dividing Right hand side with x^-2 $\frac{ x ^{-2}(x+dx)^{-2} }{ x^{-2} }$ $x^{-2} (\frac{ x+dx }{ x })^{-2}$

4. atlas

simplify it to x^-2 (1+dx/x)^-2 I hope it makes sense now

5. anonymous

I stiil dont get it Could someone please explain it step by step?

6. myininaya

Ok, do you agree that x+dx could be written as x(1+dx/x)?

7. myininaya

$x+dx=x+\frac{x}{x} dx=x+x \cdot \frac{dx}{x}=x(1+\frac{dx}{x})$

8. anonymous

how can x+dx be written as x(1+dx/x)?

9. myininaya

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

10. myininaya

Did you see what I said above what you said?

11. myininaya

|dw:1387637203504:dw|

12. myininaya

|dw:1387637215111:dw|

13. myininaya

you can factor out that x

14. anonymous

but how did you get to x+x d/x? the equation is (x+dx)^-2

15. myininaya

You wanted step by step I'm looking at the inside, x+dx, right now.

16. anonymous

ok

17. myininaya

If you can understand why x+dx=x(1+dx/x) we can move on to the exponent part

18. myininaya

19. myininaya

but i can redo

20. anonymous

i dont understand why x+dx=x(1+dx/x)

21. anonymous

where did you get the 1?

22. myininaya

Like do you the distributive property. It says: a(b+c)=ab+ac

23. anonymous

and how did you get dx/x?

24. myininaya

x(1)=x

25. myininaya

$x+dx=x+\frac{x}{x} dx$ look I multiply dx by x/x which is just one

26. myininaya

you should see an x in both terms now

27. myininaya

you can factor an x out

28. anonymous

how did you get x/x?

29. myininaya

$x(1+\frac{1}{x}dx)$ $x(1+\frac{dx}{x})$

30. myininaya

I just put it there so you would see better x/x is there if you write it because it is just one

31. myininaya

If you factor something out you are dividing the terms you factor from

32. anonymous

im sorry but i dont understnad

33. anonymous

*understand

34. myininaya

For example $3x+9=3(\frac{3x}{3}+\frac{9}{3})=3(x+3)$ Do you understand this example?

35. anonymous

Can you please write down every step you do and how you get every variable?

36. anonymous

37. myininaya

If you don't understand it separately how do you expect to understand it together?

38. myininaya

Do you understand the example I just put?

39. anonymous

no sorry

40. anonymous

im new to calculus

41. myininaya

You might want to review factoring then. Because all I did was factor 3x+9 in that example

42. anonymous

i will

43. myininaya

Do you know the distributive property?

44. anonymous

no sorry

45. myininaya

Like a(b+c) means ab+ac

46. anonymous

yeah

47. myininaya

That is the distributive property

48. anonymous

ok

49. myininaya

You can look at it either way. a(b+c)=ab+ac ab+ac=a(b+c) Do you see on this side they say hey both of these terms have an a in common so let's factor it out

50. anonymous

ok

51. myininaya

What they are doing when they factor is pulling something out (multiplying) and then also dividing by what they pulled out. Like for example , ab+ac we could factor factor out ac but that means we also need to divide each term by ac like this: $ac(\frac{ab}{ac}+\frac{ac}{ac})=ac(\frac{b}{c}+1)$

52. myininaya

What do you get when you multiply out the ac(b/c+1) expression?

53. myininaya

i must go good luck

54. anonymous

no idea

55. anonymous

thankyou

56. anonymous

I don't understand what we are supposed to do, re read your post, y +dy = (x+dx)^-2 equal = x^-2(1+dx/x)^-2 can you use the draw box below to write it again ?

57. anonymous

and dy =y'? dx =x'?

58. anonymous

$y+dy=(x+dx)^{-2} =x ^{-2}(1+\frac{ dx }{ x })^{-2}$

59. anonymous

the aim is to differentiate y=x^2

60. anonymous

2 equal signs?

61. anonymous

the second equal sign is the result

62. anonymous

once we have powered the left hand side of the equation to -2

63. anonymous

what do you mean by "ODE?"

64. anonymous

one more question: from which course , you have this problem?

65. anonymous

its from a book called calculus made easy, ill post the page i found it on right now

66. anonymous

thanks for information,

67. anonymous

no problem

68. anonymous

ill post the page in a sec

69. anonymous

70. anonymous

he adds dy or dx to expand it a bit

71. anonymous

I can understand (x +dx)^(-2) = x^(-2)(1+dx/x)^-2 but from y +dy , I cannot see how they get the right hand side. since y = x^-2 --> dy = -2x^-3 y + dy = x^-2 - 2 x^-3 = x^-2(1-2dx/x) which is not their right hand side :((

72. zpupster

this might clear it up i like this book his proofs are easier to understand than the way was taught.