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mathcalculus

  • 2 years ago

can someone explain to me Implicit Differentiation: if y^3 = 25x^2, determine dx/dt when x = 5 and dy/dt = 1

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  1. mathcalculus
    • 2 years ago
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    these are the steps but i don't understand them :/ 3y^2*dy/dt = 50x*dx/dt When x = 5, y^3 = 25*5^2 y^3 = 625 y = cube root 625 = 8.55 3(8.55)^3*1 = 50*5*dx/dt 1875 = 250dx/dt 7.5 = dx/dt

  2. mathcalculus
    • 2 years ago
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    isn't y=2.92402?

  3. mathcalculus
    • 2 years ago
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    i did this on the calculator: sqrt (625)^(1/2)

  4. mathcalculus
    • 2 years ago
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    (1/3)**

  5. mathcalculus
    • 2 years ago
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    @satellite73

  6. agent0smith
    • 2 years ago
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    Let's start with this part... \[\Large \frac{ d }{ dt } y^3\]use the chain rule to differentiate it... so bring down the exponent, reduce the exponent by one, then multiply by the derivative of y w. respect to t \[\Large 3 y^2 *\frac{ dy }{ dt }\]

  7. mathcalculus
    • 2 years ago
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    so I can't replace the dy/dt by 1 since it's given?

  8. mathcalculus
    • 2 years ago
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    @agent0smith

  9. agent0smith
    • 2 years ago
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    Well yes but i was seeing if you understood the differentiation... don't worry about plugging in numbers till later.

  10. mathcalculus
    • 2 years ago
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    and 3y^2 is already the derivative, why do we need to use the chain rule?

  11. mathcalculus
    • 2 years ago
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    @agent0smith

  12. agent0smith
    • 2 years ago
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    Because you have to multiply by the derivative of y with respect to t.

  13. agent0smith
    • 2 years ago
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    3y^2 is NOT the derivative of y^3, UNLESS you're just differentiating with respect to y.

  14. tpmys
    • 2 years ago
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    normally we have y dependent on x here y, and x both are dependent on some t, so we dont know what the function is so instead of y^3 >>> 3y^2 we have the chain rule y^3 >>> 3y^2 * y'

  15. agent0smith
    • 2 years ago
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    It's the same with dy/dx... the derivative of y with respect to x is not just 1... it's 1*dy/dx

  16. mathcalculus
    • 2 years ago
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    ok... :/ it's a little confusing..

  17. ranga
    • 2 years ago
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    agent0smith explained it nicely: \[\frac{ d }{ dy }(y ^{3}) =3y ^{2}\] But \[\frac{ d }{ dt }y ^{3} = \frac{ d }{ dy }y ^{3}\frac{ dy }{ dt } = 3y ^{2}\frac{ dy }{ dt }\]

  18. agent0smith
    • 2 years ago
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    It's the same thing with the chain rule \[\huge \frac{ d }{ dx } f(x)^n = n*f(x)^{n-1} * f \prime (x)\] you have to always multiply by the derivative at the end.

  19. mathcalculus
    • 2 years ago
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    why can't we plug in the numbers right away if it's given?

  20. agent0smith
    • 2 years ago
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    You can... you're probably just better off not doing it until you really know what you're doing.

  21. mathcalculus
    • 2 years ago
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    right i understand that... @ranga

  22. mathcalculus
    • 2 years ago
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    well i was taught to plug in.. that's why.

  23. mathcalculus
    • 2 years ago
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    that's the confusing part.

  24. agent0smith
    • 2 years ago
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    Fair enough... but you can't plug in until you have this step: 3y^2*dy/dt = 50x*dx/dt

  25. mathcalculus
    • 2 years ago
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    exactly what i have right now.

  26. mathcalculus
    • 2 years ago
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    so dy/dt is 1...

  27. mathcalculus
    • 2 years ago
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    in the steps above, why is this: 3(8.55)^3*1 = 50*5*dx/dt?

  28. mathcalculus
    • 2 years ago
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    y=8.55? y is suppose to be 2.92402.

  29. mathcalculus
    • 2 years ago
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    and it's raised to the 3rd, and why not 2?

  30. agent0smith
    • 2 years ago
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    When x = 5, y^3 = 25*5^2 y^3 = 625 y = cube root 625 = 8.55 i think the next line after that is a mistake and it should by to the power of 2

  31. mathcalculus
    • 2 years ago
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    so y=2.92402 is correct?

  32. agent0smith
    • 2 years ago
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    y^3 = 625 y = cube root 625 = 8.55

  33. mathcalculus
    • 2 years ago
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    how?

  34. mathcalculus
    • 2 years ago
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    i did this in the calculator: sort(625)^(1/3)

  35. mathcalculus
    • 2 years ago
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    sqrt*

  36. agent0smith
    • 2 years ago
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    And you'll get 8.55... 2.94 cubed is close to 3^3 which is 27. Not 625.

  37. mathcalculus
    • 2 years ago
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    i swear im getting 2.92402

  38. mathcalculus
    • 2 years ago
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    i plugged it in exactly like iwrote it.

  39. agent0smith
    • 2 years ago
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    sort(625)^(1/3) why are you taking the square root of a cubed root...

  40. agent0smith
    • 2 years ago
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    y^3 = 625 y = 625^(1/3)

  41. mathcalculus
    • 2 years ago
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    OHHH

  42. mathcalculus
    • 2 years ago
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    i thought the sqrt could do cube root.

  43. mathcalculus
    • 2 years ago
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    if we did the exponent also.

  44. agent0smith
    • 2 years ago
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    That would mean ((625^(1/3))^(1/2) = 625^(1/6)

  45. mathcalculus
    • 2 years ago
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    thank you! i got it !

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