What is the constant $c$ in $\frac{d}{dx} \frac{d}{dy} c^{xy} = c^{xy}$?

Question

In the manner in which $\frac{d}{dx} e^{x} = e^{x}$. What is the value of the constant $c$ for which $\frac{d}{dx} \frac{d}{dy} c^{xy} = c^{xy} $ ?

Answer 1

Assuming $c\ne0$

$$ \frac{\rm d}{{\rm d}y} c^{xy} = xc^{xy}\ln(c)$$

$$\frac{\rm d}{{\rm d}x} \left(xc^{xy}\ln(c)\right) = xy\ln^2(c)c^{xy}+c^{xy}\ln(c) =c^{xy}$$

Cancelling $c^{xy}$

$$xy\ln^2(c)+\ln(c)=1\implies xy=\frac{1-\ln(c)}{\ln^2(c)}$$assuming $c\ne1$

Clearly this cannot hold for all $x$ and $y$ since the left expression is constant.

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Now we ignored 2 values of $c$, i.e, $0$ and $1$.

Checking $c=1$ this doesn't satisfy, since $\frac{\rm d}{{\rm d}y}(1)=0\ne 1$

Checking $c=0$ this does satisfy.

Thus the only solution is $c=0$