# dy/dx

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

1. ### 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
2. ### 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

MathLover1, Sep 20, 2021
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3. ### nycmathguy

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Cool. I can't wait for Calculus. Excited.

nycmathguy, Sep 20, 2021
4. ### 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. ### 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
6. ### HallsofIvy

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I wasn't using Latex because the original post did not use Latex.

HallsofIvy, Nov 9, 2021
7. ### nycmathguy

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nycmathguy, Nov 9, 2021
8. ### HallsofIvy

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I have tried $$x^2$$, $x^2$, and $x^2$. None seem to work. What should I do?

HallsofIvy, Nov 9, 2021
9. ### MathLover1

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MathLover1, Nov 9, 2021
10. ### 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
11. ### MathLover1

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that is true, but to convert to LaTex form is not a point

MathLover1, Nov 10, 2021
12. ### nycmathguy

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Do you know a good app or site that converts to LaTex?

nycmathguy, Nov 10, 2021
13. ### MathLover1

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https://mathpix.com/

Snip can convert images into LaTeX for inline equations, block mode equations, and numbered equations. Snip also supports some text mode LaTeX, like the tabular environment.

scroll down and watch videos

but it's not free for everyone, only for students

Up to 50 Snips
Up to 20 PDF pages

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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