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Math Help - Derivative

  1. #1
    Member classicstrings's Avatar
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    Derivative

    Find the 2nd derivative of tan(2x).

    The first is sec^2(2x). Then you would change it to 1/(cos^2(2x)). How would you go from there? Thanks
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  2. #2
    Senior Member ecMathGeek's Avatar
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    Quote Originally Posted by classicstrings View Post
    Find the 2nd derivative of tan(2x).

    The first is sec^2(2x). Then you would change it to 1/(cos^2(2x)). How would you go from there? Thanks
    If you already know how to find the derivative of sec(2x), you don't need to change sec^2(2x) into 1/cos^2(2x).

    I will go about this assuming you know how to take the derivative of secant.

    f(x) = tan(2x)

    The derivative of f(x) involves the chain rule (we need to use it twice):
    f'(x) = sec^2(2x)*2 <-- the derivative of tan(2x) times the derivative of 2x.
    f'(x) = 2sec^2(2x)

    The derivative of f'(x) involves the chain rule (we need to use it three times):
    f''(x) = 2*2sec(2x)*sec(2x)tan(2x)*2 <-- 2 times the derivatives of sec^2(2x) times the derivative of sec(2x) times the derivative of 2x.
    f''(x) = 8sec^2(2x)tan(2x)
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    Quote Originally Posted by classicstrings View Post
    Find the 2nd derivative of tan(2x).

    The first is sec^2(2x). Then you would change it to 1/(cos^2(2x)). How would you go from there? Thanks
    No the first is,
    2*sec^2(2x)
    By chain rule.

    But you do not need to change it.

    You can write, by Pythagoren identity,

    2*(1+tan^2(2x)) = 2 + 2*tan^2(2x)

    You you can differenciate.
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