## mathcalculus one year ago Find the slope of the tangent line to the curve (a lemniscate) 2(x^2+y^2)^2=25(x^2-y^2) at the point (-3,1)?

1. hartnn

could you implicitly differentiate that ?

2. hartnn

you will need chain rule here whats d/dx x^2 =...? whats d/dx y^2=...?

3. hartnn

what ? where does 2*2 come from ?

4. hartnn

d/dx[2(x^2+y^2)^2] = 2*(2 (x^2+y^2)) times d/dx(x^2+y^2) this is because of chain rule, got this step ?

5. hartnn

d/dx (x^2) = 2x thats why and the other 2 was already there

6. hartnn

ok, so now what about d/dx (x^2+y^2) =...?

7. mathcalculus

2x+2y?

8. hartnn

here, chain rule again comes into picture d/dx f(y) = f'(y) dy/dx so d/dx (y^2) = 2y dy/dx got this ?

9. hartnn

if you got that we are almost done :)

10. hartnn

d/dx[2(x^2+y^2)^2] = 2*(2 (x^2+y^2)) times d/dx(x^2+y^2) this part ?

11. hartnn

whats d/dx x^2 =.... ?

12. hartnn

we don't rewrite anything

13. mathcalculus

ohman :/

14. hartnn

derivative of outer times derivative of inner like for (sin x)^3 we have outer = x^3, derivative = 3x^2 inner = sin x derivative = cos x so, 3 (sin x)^2 cos x got this ?

15. hartnn

in same way d/dx [(x^2+y^2)^2] = 2 (x^2+y^2) times d/dx (x^2+y^2)

16. mathcalculus

i get that, but in this problem im confused with why the exponent goes away

17. hartnn

ok, you said d/dx (x^2) = 2x why did the exponent go away here ?

18. mathcalculus

2(x^2+y^2)^2=25(x^2-y^2)

19. mathcalculus

so you're finding the derivative of the inside?

20. hartnn

we do find derivative of inside in chain rule, right ?

21. mathcalculus

yup

22. mathcalculus

my professor also taught us a way of outer first then rewrite inner and then multiple by the derivative of the inner.

23. hartnn

he must have meant "derivative of outer first ....."

24. mathcalculus

d/dx[f(g(x))]= f' (g(x))*g'(x)

25. hartnn

yes, here f(x) was x2

26. mathcalculus

well isn't it 4?

27. mathcalculus

2(x^2+y^2)^2 |dw:1381774415672:dw|

28. hartnn

yes, it is 4.

29. mathcalculus

oh so then it is. where are we getting x2?

30. hartnn

d/dx[2(x^2+y^2)^2] = 2*(2 (x^2+y^2)) times d/dx(x^2+y^2) = 4(x^2+y^2)times d/dx(x^2+y^2)

31. hartnn

sorry i didn't get your question?

32. mathcalculus

got that =]

33. mathcalculus

so can it be written like this? : d/dx 4(x^2+y^2) (2x+2y) ?

34. hartnn

no, why would d/dx come in front again ?

35. mathcalculus

so i get rid of it?

36. mathcalculus

since i have the derivative already?

37. hartnn

d/dx [(x^2+y^2)^2] = 2 (x^2+y^2) times d/dx (x^2+y^2) = 4 (x^2+y^2) times (2x+2y dy/dx) see whether you get this?

38. mathcalculus

yeah i got it, but why don't we do dy/dx here: 4 (x^2+y^2) times (2x+2y dy/dx)|dw:1381774790286:dw|

39. mathcalculus

we can't bc we have to leave it as it is right? according to the chain rule

40. hartnn

because there, you have not differentiated x^2+y^2 yrt

41. hartnn

*yet

42. hartnn

and yes, we need to leave it as it is, that too

43. mathcalculus

ok

44. mathcalculus

now what do we do

45. mathcalculus

same for the other side?

46. hartnn

yes, other side is too simple. try ?

47. hartnn

right side is just 25(2x-2y dy/dx) isn't it ?

48. mathcalculus

25(x^2-y^2)*(2x-2y*dy/dx)

49. hartnn

what why ? we need chain rule for just y^2 here, isn't it ?

50. mathcalculus

huh i did?

51. hartnn

its like 25x^2 - 25y^2 derivative = 25(2x) - 25 (2y dy/dx)

52. mathcalculus

where did i go wrong?

53. hartnn

the x^2-y^2 part was unnecessary, you though it was chain rule again ?

54. mathcalculus

wait it wasn't because it had no exponent?

55. hartnn

yes, correct.

56. mathcalculus

25*(2x-2y*dy/dx)

57. mathcalculus

so its this?

58. mathcalculus

got cha.

59. mathcalculus

now do we plug in x and y?

60. hartnn

correct! put x=-3, y=1 in that...

61. hartnn

i assume you are simplifying...

62. mathcalculus

i don't know i keep getting the answer wrong whenever i try to submit it

63. hartnn

what did u get for dy/dx ?

64. mathcalculus

i keep getting incorrects :/

65. mathcalculus

i got 3/119

66. hartnn

4 (x^2+y^2) times (2x+2y dy/dx) = 25(2x-2y dy/dx) x=-3, y=1 4(9+1) (-6 +2 dy/dx) = 25 (-6 -2 dy/dx)

67. mathcalculus

ohi wrote one of the signs wrong.

68. mathcalculus

got it.

69. mathcalculus

thank you so much!

70. mathcalculus

:)

71. hartnn

welcome ^_^