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.
So,
$ 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, $$\frac{dy}{dx} = 1$$
