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Thread: Implicit Differentiation

  1. #1
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    Implicit Differentiation

    Given
    $\displaystyle x^2 + 25y^2 = 100$
    show that $\displaystyle \frac{d^2y}{dx^2} = - \frac{4}{25y^3}$

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  2. #2
    TD!
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    Differentiate and solve for y':

    $\displaystyle
    x^2 + 25y^2 = 100 \Rightarrow 2x + 50yy' = 0 \Leftrightarrow y' = - \frac{x}{{25y}}
    $

    Differentiate again and substitute y' with the previous expression:

    $\displaystyle
    y'' = \frac{{xy' - y}}{{25y^2 }} = \frac{{x\left( { - \frac{x}{{25y}}} \right) - y}}{{25y^2 }} = \frac{{ - \frac{{x^2 + 25y^2 }}{{25y}}}}{{25y^2 }}
    $

    Now you can use the initial relation again in the numerator:

    $\displaystyle
    y'' = \frac{{ - \frac{{x^2 + 25y^2 }}{{25y}}}}{{25y^2 }} = \frac{{ - \frac{{100}}{{25y}}}}{{25y^2 }} = \frac{{ - \frac{4}{y}}}{{25y^2 }} = \boxed{- \frac{4}{{25y^3 }}}
    $
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  3. #3
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    Hello, ^_^Engineer_Adam^_^!

    Given: .[1] $\displaystyle x^2 + 25y^2 = 100,$ .show that: .$\displaystyle \frac{d^2y}{dx^2} = - \frac{4}{25y^3}$

    Differentiate: .$\displaystyle 2x + 50y\left(\frac{dy}{dx}\right) \:=\:0\quad\Rightarrow\quad\frac{dy}{dx}\:=\:-\frac{1}{25}\,\frac{x}{y}$ [2]

    Differentiate again: .$\displaystyle \frac{d^2y}{dx^2} \:=\:-\frac{1}{25}\left[\frac{y - x(\frac{dy}{dx})}{y^2}\right] $

    Substitute [2]: .$\displaystyle \frac{d^2y}{dx^2} \:=\:-\frac{1}{25}\left[\frac{y - x(-\frac{x}{25y})}{y^2}\right] \:=\:-\frac{1}{25}\left[\frac{y + \frac{x^2}{25y}}{y^2}\right]$

    Multiply top and bottom by $\displaystyle 25y\!:\;\;\frac{d^2y}{dx^2} \:=\:-\frac{1}{25}\left[\frac{25y^2 + x^2}{25y^3}\right]$

    From [1], the numerator equals 100: .$\displaystyle \frac{d^2y}{dx^2} \:=\:-\frac{1}{25}\left[\frac{100}{25y^3}\right]$

    Therefore: .$\displaystyle \boxed{\frac{d^2y}{dx^2} \:=\:-\frac{4}{25y^3}}$

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