I'm given that $$ x^{{mx}^{mx}...} = y^{{my}^{my}...}$$

I should find $ \frac {dy}{dx} $.

How do I start? Is there any way to simplify this? For example, do the extra exponents stop mattering after some point (in the same, can we say something like: this is only true if $ x = y $? (I'm just guessing -- I'm not really claiming that this is the case).


As it is an infinite series, inserting of deleting a term doesn't make any difference.


$ x^{{mx}^{mx}...} = y^{{my}^{my}...}$

$ (x^{{m)x}^{mx}...} = (y^{{m)y}^{my}...}$

Taking $log$ on both sides

$log[(x^{{m)x}^{mx}...}] = log[(y^{{m)y}^{my}...}]$

$x^{mx^{mx...}}log[x^m] = y^{my^{my...}}log[y^m]$

As, $x^{{mx}^{mx}...} = y^{{my}^{my}...}$, cancelling it from both sides,

$log[x^m] = log[y^m]$

$mlog(x) = mlog(y)$

$log(x) = log(y) $

$x = y$

Thus, $$\frac{dy}{dx} = 1$$


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