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Math Help - differentiate

  1. #1
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    differentiate

    (1+cosx) / (1-cosx)
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  2. #2
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    Solution

    By using the division rule of differentiation:

    d/dx(f(x)/g(x)) = f'(x)*g(x) - g'(x)*f(x)/(g(x))^2

    So, the derivative of 1 + cos(x) / 1 - cos(x) looks like:

    [d/dx(1+cos(x)) * (1-cos(x)) - d/dx(1 - cos(x)) * (1 + cos(x))] / (1-cos(x))^2

    which simplifies to:

    [-sin(x)*(1 - cos(x)) - sin(x)*(1 + cos(x)) ] / (1-cos(x))^2

    which further simplifies to:

    -sin(x) +sin(x)cos(x) - sin(x) - sin(x)cos(x) / (1 - cos(x))^2

    which further simplifies to:

    -2sin(x) / (1 - cos(x))^2

    I'm not sure if this can be further simplified. Perhaps you could expand the denominator and do some additional canceling or condensing through the use of trigonometric identities.

    I hope this is helpful.
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