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Math Help - Derivatives of Inv. Trig

  1. #1
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    Derivatives of Inv. Trig

    Hello everyone. I need help on the following problem involving inverse trig.

    A) Show that the derivative of ( arccot[x] - arctan[1/x] ) = 0 for all x not equal to 0.

    B) Prove that there is no constant C such that arccot[x] - arctan[1/x] = C for all x not equal to 0. Explain why this does not contradict the zero-derivative theorem.

    Any suggestions will be appreciated
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by RB06 View Post
    Hello everyone. I need help on the following problem involving inverse trig.

    A) Show that the derivative of ( arccot[x] - arctan[1/x] ) = 0 for all x not equal to 0.

    B) Prove that there is no constant C such that arccot[x] - arctan[1/x] = C for all x not equal to 0. Explain why this does not contradict the zero-derivative theorem.

    Any suggestions will be appreciated
    Let x = tan(y), then 1/x = cot(y). Hence (assuming that atan and acot
    return values in the range (-pi/2, pi/2):

    y= atan(x) = acot(1/x).

    So: acot(x) - atan(1/x) = 0, and so the derivative is zero (the derivative of a constant function is zero).

    Also if atan abd acos have the restriction on range we have also proven part B.

    RonL
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  3. #3
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    Hello, RB06!

    A) Show that the derivative of: .f(x) .= .arccot(x) - arctan(1/x) .
    . . .equals 0 for all x ≠ 0.
    How about differentiating?

    We have: .f(x) .= .arccot(x) - arctan(x^{-1})

    . . . . . . . .-1 . . . . . -x^{-2}
    f'(x) .= . -------- - ---------------
    . . . . . . .1 + x . .1 + x^{-2}


    Multiply top and bottom of the second fraction by x:

    . . . . . . . .-1 . . . . . 1
    f'(x) .= .-------- + -------- . = . 0
    . . . . . . 1 + x - -x + 1

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