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Math Help - To differentiate a trig func. using the logarithms, need your opin's on my solution

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    Member dokrbb's Avatar
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    To differentiate a trig func. using the logarithms, need your opin's on my solution

    Sorry for not using LaTeX, I'm not used to it yet, but I promise I would improve, so the equation is:

    Let f(x)= 6tan^(-1)(4e^(x). Need to find the derivative


    I continued to solve the equation, before differentiating f(x) = y = 6(1/(tan(4e^(x))) = 6cot (4e^(x)),

    Now, ln y = ln (6cot (4e^(x))) =>
    ln y = ln 6 + ln(cot(4e^(x))) =>

    y/y = (1/6) + (1/ cot(4e^(x))) *(- csc^(2) (4e^(x)))*(4xe^(x-1))) =>

    y/y = (1/6) - [(4xe^(x-1))*(csc^(2)(4e^(x)]/[cot(4e^(x)], and =>

    y = 6cot (4e^(x)) * {(1/6) - [(4xe^(x-1))*(csc^(2)(4e^(x))]/[(cot(4e^(x))]},

    I definitely have some problems with log functions, I think...

    Thank you
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    Re: To differentiate a trig func. using the logarithms, need your opin's on my soluti

    Quote Originally Posted by dokrbb View Post
    Sorry for not using LaTeX, I'm not used to it yet, but I promise I would improve, so the equation is:

    Let f(x)= 6tan^(-1)(4e^(x). Need to find the derivative
    Is it possible that \tan^{-1} really is the inverse tangent \arctan~?
    Thanks from dokrbb
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    Member dokrbb's Avatar
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    Re: To differentiate a trig func. using the logarithms, need your opin's on my soluti

    do you mean that I had to do in this way:

    f(x) = y = 6(arctan(4e^(x))),

    Now, ln y = ln (6arctan (4e^(x))) =>
    ln y = ln 6 + ln(arctan(4e^(x))) =>

    y/y = (1/6) + (1/ (1 + (4e^(x))^(2)) * (4xe^(x-1)))

    y/y = (1/6) + (4xe^(x-1))/(1 + (16e^(2x)), and

    y = 6(arctan(4e^(x))* {(1/6) - [(4xe^(x-1))]/[(1 + (16e^(2x))]},

    (I wonder, do I have to rewrite and solve for ln y = ln 6 + x ln(arctan(4e), ot that doesn't change the result )

    y/y = (1/6) + ln(arctan(4e) + x*[1/ (1 + (4e^(x))^(2))] * (4e) =>


    y = 6(arctan(4e^(x))* {(1/6) + ln(arctan(4e) + [(4xe)]/[(1 + (16e^(2x))]},


    well. I just tried to solve by two ways, and now I'm not sure whether both of them is wrong or only the first one...
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    Re: To differentiate a trig func. using the logarithms, need your opin's on my soluti

    Quote Originally Posted by dokrbb View Post
    do you mean that I had to do in this way:
    f(x) = y = 6(arctan(4e^(x))),
    If it is indeed y = 6\arctan(4e^x), why in the world are you doing anything with logarithms?

    y^{\prime}=\frac{24e^x}{1+16e^{2x}}
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    Member dokrbb's Avatar
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    Re: To differentiate a trig func. using the logarithms, need your opin's on my soluti

    this would sound as a dumb answer, but it's a homework set of problems after discussing the "Logarithmic differentiation", the teacher's argument was that sometimes, when we have some complex functions, would be easier (sounds funny after comparing your answer with mine ) to use log differentiation; Like I said - is a dumb answer, but I assumed they want us to apply the Log differentiation for this chapter ...
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    Re: To differentiate a trig func. using the logarithms, need your opin's on my soluti

    Quote Originally Posted by dokrbb View Post
    this would sound as a dumb answer, but it's a homework set of problems after discussing the "Logarithmic differentiation", the teacher's argument was that sometimes, when we have some complex functions, would be easier (sounds funny after comparing your answer with mine ) to use log differentiation; Like I said - is a dumb answer, but I assumed they want us to apply the Log differentiation for this chapter ...

    Quite often textbooks will throw in a question that has nothing to do with the current topic. It is a way of review and keeps students on their toes.
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