hooverst 3 years ago y=e^xlnx y=ln(3x^2) y=lnx/x^2+1

1. dpaInc

what's the question? which equation is nicest?

2. dpflan

What's the goal? Solve for x in this system of equations?

3. dpflan

heh, I like the first one ;)

4. dpaInc

me too! :)

5. hooverst

each one is a different question its says compute the derivatives of each function

6. dpflan

Oh, OK, are you familiar with (1.) product rule and (2.) quotient rule and the (3.) power rule?

7. hooverst

yeah a little

8. dpflan

(1.) $y=e^x*ln(x)$ (2.) $y=ln(3x^2)$ (3.) $y=\frac{ln(x)}{x^2}+1$

9. dpflan

And you know chain rule? a little, right? ;)

10. hooverst

the 1st equation is $y=e ^{xlnx}$

11. hooverst

yes

12. dpflan

OK, well, let's work with equation 2, for me that is the easiest one. We are differentiating with respect to y, right?

13. hooverst

yes

14. dpflan

Cool, $y=ln(3*x^2)$ We will use the Chain Rule here

15. dpflan

$\frac{d}{dx}(ln(3*x^2)) = \frac{d*ln(u)}{du}*\frac{du}{dx}$ $u = 3*x^2$and $\frac{d*ln(u)}{du} = \frac{1}{u}$ $\frac{du}{dx} = 3*2*x^{2-1} = 6x$

16. dpflan

Yes, and to get du/dx we use the power rule

17. dpflan

$x^y = y*x^{y-1}$ is the power rule

18. dpflan

Can you finish solving the equation?

19. hooverst

ok cause when he explained it in class i did not understand it at all.

20. dpflan

All right, how is it now?

21. hooverst

ok so for the power rule what do i plug in or do I have to plug anything in

22. dpflan

Nah, you don't need to plug-in anything. So, what do you have so far? If you are writing it down, you can snap a picture with your phone if it has a camera, then post the picture here

23. dpflan

$\frac{d}{dx}(ln(3*x^2)) = \frac{d*ln(u)}{du}*\frac{du}{dx}$ $\frac{d*ln(u)}{du} = \frac{1}{u}$ $\frac{du}{dx} = 3*2*x^{2-1} = 6x$ $u=3*x^2$ So you have $\frac{1}{3*x^2} * 6x$ Simplify and you're done with that equation

24. hooverst

O ok I didn't know whether what I had was right but thats what I just got to.

25. dpflan

This maybe the hardest of the bunch because of the substitution, chain rule, and product rule

26. hooverst

yea It was

27. dpflan

Well, because substitution can be confusing because you introduce a new variable to differentiate on

28. dpflan

*or with respect to

29. hooverst

O ok

30. dpflan

Now (1.)$y=e^x * lnx$

31. Callisto

Excuse ... me.... are we finding dy/dx or dx/dy?

32. dpflan

dy/dx I believe

33. Callisto

34. dpflan

Heh, thanks, are you following along @Callisto ?

35. dpflan

$y=e^x* lnx$ Here we use the product rule. We can think of this equation as the product of 2 functions of x

36. dpflan

$(f*g)' = f'*g + f*g'$ So the derivative of the product of these functions, f and g Is the sum of the derivative of the first * the second + the first * the derivative of the second

37. Callisto

For the second one, I was thinking in this way.. $y=ln(3x^2)=ln3+ lnx^2 = ln3+2lnx$$\frac{dy}{dx}=\frac{d}{dx}2lnx = ...$

38. dpflan

Yes, that is easier! @hooverst If you did it that way you don't have do substitution

39. hooverst

Ok dpflan

40. dpflan

It can be really fun in math the find the simpler way of doing things, so you just use some intuition and understanding to make your work easier, simpler can be much more beautiful too

41. dpflan

*to find

42. dpflan

Nice one @Callisto @hooverst So, you think you can solve equation (1.) with the product rule?

43. dpaInc

hold on... i think @hooverst clarified that the first equation is: $$\large y=e^{xlnx}$$ , not $$\large y=e^{x}\cdot lnx$$

44. dpflan

Heh, you're right, man

45. hooverst

so callisto when you solved your equation for the second 1 what did you get for your anserw

46. dpflan

Thanks, getting a little carried away. Let me step away for a bit

47. dpaInc

ur doing fine....:)

48. Callisto

@hooverst What did you get ??

49. hooverst

is the anserw $y=\ln(3x^2)=1\div 3x^2$

50. Callisto

Not really... what is $$\frac{d}{dx}lnx$$ ?

51. hooverst

im not sure

52. dpaInc

are we still working on the second equation? @hooverst ???

53. dpaInc

because you can do this many different ways... but in the end, the derivative should be the same...

54. hooverst

yes but I was trying to figure out was 2lnx the anserw for #2 or is y=ln(3x^2)=1/3x^2 the anserw

55. dpaInc

answer as in the derivative? you and @dpflan , worked it out to $$\large \frac{6x}{3x^2}$$ simplified to....???

56. dpaInc

in the case with @Callisto , her method was to simplify the function first: $$\large y=ln(3x^2)=ln3+2lnx$$ so $$\large y'=[ln3]'+[2lnx]'=$$ ...???

57. dpaInc

and either way the derivative is the same...

58. Callisto

The key point is you need to know what $$\frac{d}{dx}lnx$$ is.

59. dpflan

Yes, that is a derivative you need to memorize, it will be quite useful

60. hooverst

Ok so many of you have said had your own opinion about the equation so which one is the right 1.

61. dpflan

Hehe, right, so, what is your opinion? ;p You can solve a problem many, many, different ways

62. dpflan

For you, how would you approach it now that you've seen how we would?

63. Callisto

I would say, none of us have given you the final answer. But we have given you some steps you need using different approaches. Though, the final answer will be the same.

64. hooverst

Ok Think the one Callisto gave me is more simple for #2

65. dpflan

Definitely, that was an awesome application of intuition

66. dpflan

well, and mathematical understanding

67. dpaInc

i guess it comes down to preference because i personally would've used u'/u ... the method used earlier...

68. Callisto

The step I left for you is to find $$\frac{d}{dx}lnx$$. Multiply the answer you get for that by 2. Then, it's done.

69. hooverst

ok

70. Callisto

May I ask you again- what is d/dx ( lnx) ?

71. hooverst

would you multiply it.

72. Callisto

Nope.... Hint: look it up in the pdf.: http://tutorial.math.lamar.edu/pdf/Calculus_Cheat_Sheet_Derivatives.pdf You can find the answer for d/dx (lnx) there.

73. hooverst

Ok is it d/dx(ln(x))=1/x

74. Callisto

Yes. dy/dx = d/dx (ln3 + lnx^2) = d/dx (2lnx) = 2 d/dx(lnx) = ...?

75. hooverst

is it 2/x or just 2x Idon't know if its right

76. Callisto

Which one do you think? (i) 2/x (ii) 2x

77. hooverst

2/x

78. Callisto

Yes. That's correct. Any questions?

79. hooverst

just 1 when we started simplifying the equation where did you get the 2 from?

80. Callisto

$lnx^a = a\ln x$

81. hooverst

Ok I get it. thanks Callisto

82. Callisto

Welcome.