# Differentiation equations with a power tower

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