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Math Help - Derivative of inverse tan fuction

  1. #1
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    Derivative of inverse tan fuction

    I was having trouble with the following question about finding the derivative of:

    y=(tan^-1(x/2))^4

    -i utilized the chain rule but this came out to be wrong.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by johnsy123 View Post
    I was having trouble with the following question about finding the derivative of:

    y=(tan^-1(x/2))^4

    -i utilized the chain rule but this came out to be wrong.
    Please show us your work; it will make it easier to pinpoint the error you made!
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  3. #3
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    \displaystyle \begin{align*} y &= \left[\arctan{\left(\frac{x}{2}\right)}\right]^4 \\ y^{\frac{1}{4}} &= \arctan{\left(\frac{x}{2}\right)} \\ \tan{\left(y^{\frac{1}{4}}\right)} &= \frac{x}{2} \\ \frac{d}{dx}\left[\tan{\left(y^{\frac{1}{4}}\right)}\right] &= \frac{d}{dx}\left(\frac{x}{2}\right) \\ \frac{d}{dy}\left[\tan{\left(y^{\frac{1}{4}}\right)}\right]\,\frac{dy}{dx} &= \frac{1}{2} \\ \frac{1}{4}y^{-\frac{3}{4}}\sec^2{\left(y^{\frac{1}{4}}\right)}\,  \frac{dy}{dx} &= \frac{1}{2} \\ \frac{dy}{dx} &= 2y^{\frac{3}{4}}\cos^2{\left(y^{\frac{1}{4}}\right  )}\end{align*}

    You could convert it back to a function of \displaystyle x now if you want.
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