dy/dx

Discussion in 'Differentiation and Integration' started by LoL Maths, Sep 20, 2021.

  1. LoL Maths

    LoL Maths

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    can anyone please solve it : (x^m)^(x^m)^(x^m)=(y^n)^(y^n)^(y^n) find dy/dx
     
    LoL Maths, Sep 20, 2021
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  2. LoL Maths

    MathLover1

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    (x^m)^((x^m)^(x^m))=(y^n)^((y^n)^(y^n))

    (y^n)^((y^n)^(y^n))-(x^m)^((x^m)^(x^m))=0


    [​IMG]
     
    MathLover1, Sep 20, 2021
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  3. LoL Maths

    nycmathguy

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    Cool. I can't wait for Calculus. Excited.
     
    nycmathguy, Sep 20, 2021
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  4. LoL Maths

    HallsofIvy

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    Frankly, I don't understand MathLover1's answer! I do understand that MathLover1 changed the problem slightly. (x^m)^(x^m)^(x^m) is ambiguous. It could be either [(x^m)^(x^m)]^(x^m), which is what MathLover1 changed it to, or it could be (x^m)^[(x^m)^(x^m)], which is much more difficult.

    For example, 2^3^2 is ambiguous. It could be (2^3)^2= 8^2= 64[ or it could be 2^{3^2}= 2^9= 1024.


    (x^a)^b= x^{ab} so (x^m)^(x^m)= x^{mx^m} and
    [(x^m)^(x^m)]^(x^m)= (x^{mx^m})^{x^m}= x^{m(x^m)(x^m)}= x^{mx^{2m}}.

    To differentiate that, let \(u= mx^{2m}\) and we have \(x^u\).

    Since u is a function of x, we need to use "logarithmic differentiation". Let \(y= x^u\). Then \(ln(y)= ln(x^u)= u ln(x)\). The derivative of \(ln(y)\) with respect to x is \(\frac{1}{y}\frac{dy}{dx}\).

    On the right, use the product rule. \(\frac{duln(x)}{dx}= \frac{du}{dx}ln(x)+ u\frac{dln(x)}{dx}\).
    Since \(u= mx^{2m}\), \(\frac{du}{dx}= 2m^2x^{2m- 1}\). Of course \(\frac{dln(x)}{dx}= \frac{1}{x}\). So \(\frac{duln(x)}{dx}= 2m^2x^{2m- 1}ln(x)+ mx^{2m}/x= x^{2m-1}(2m^2ln(x)+ m)\).

    So we have \(\frac{1}{y}\frac{dy}{dx}= x^{2m-1}(2m^2ln(x)+ m\).

    Since \(y= x^u= x^{mx^{2m}}\)
    \(\fra{dy}{dx}= x^{mx^{2m}+ 2m- 1}(2m^2ln(x)+ m))\).
     
    Last edited: Nov 8, 2021
    HallsofIvy, Nov 8, 2021
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  5. LoL Maths

    nycmathguy

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    Your LaTex is not appearing as LaTex. Take a look at your replies involving this style of typing.
     
    nycmathguy, Nov 8, 2021
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  6. LoL Maths

    HallsofIvy

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    I wasn't using Latex because the original post did not use Latex.
     
    HallsofIvy, Nov 9, 2021
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  7. LoL Maths

    nycmathguy

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    I cannot read what you typed. Take a look at your reply.
     
    nycmathguy, Nov 9, 2021
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  8. LoL Maths

    HallsofIvy

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    I have tried \(x^2\), [math]x^2[/math], and $x^2$. None seem to work. What should I do?
     
    HallsofIvy, Nov 9, 2021
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  9. LoL Maths

    MathLover1

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    MathLover1, Nov 9, 2021
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  10. LoL Maths

    nycmathguy

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    Wolfram does not provide a screen for typing math solutions to convert to LaTex form. The tiny space bar allows for typing certain questions and, of course, most equations.
     
    nycmathguy, Nov 10, 2021
    #10
  11. LoL Maths

    MathLover1

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    that is true, but to convert to LaTex form is not a point
     
    MathLover1, Nov 10, 2021
    #11
  12. LoL Maths

    nycmathguy

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    Do you know a good app or site that converts to LaTex?
     
    nycmathguy, Nov 10, 2021
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  13. LoL Maths

    MathLover1

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    https://mathpix.com/

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    here are 3 Free Word To LaTeX Converter Software For Windows

    https://www.ilovefreesoftware.com/15/featured/free-word-to-latex-converter-software-windows.html
     
    Last edited: Nov 10, 2021
    MathLover1, Nov 10, 2021
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  14. LoL Maths

    nycmathguy

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    I will visit the links provided.
     
    nycmathguy, Nov 10, 2021
    #14
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