dy/dx

(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


MSP189424329e54b6c21ha300001i62g80hh7723123
 
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:
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 \(y= x^u= x^{mx^{2m}}\)
\(\fra{dy}{dx}= x^{mx^{2m}+ 2m- 1}(2m^2ln(x)+ m))\).

Your LaTex is not appearing as LaTex. Take a look at your replies involving this style of typing.
 
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.

that is true, but to convert to LaTex form is not a point
 
Do you know a good app or site that converts to LaTex?


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
Sign up with your school email (.edu) and get more free Snips and PDFs!

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:
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
Sign up with your school email (.edu) and get more free Snips and PDFs!

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

I will visit the links provided.
 

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