# Numerically stable reverse automatic differentiation of power(x, y)?

I would like to compute the adjoints $$\bar x$$ and $$\bar y$$, from a reverse automatic differentiation perspective, of the expression $$x^y$$. The adjoint $$\bar{x^y}$$ is already known; and we can assume $$x \geq 0$$ and $$y \geq 1$$.

The "easy" solution consists of rewriting the expression $$x^y=e^{y \ln x}$$, and proceed piece-wise from there. However, this solution proves to be unstable numerically, it gives:

$$\bar y = \bar{x^y} x^y \ln x$$

and

$$\bar x = \bar{x^y} x^{y-1} y$$

Is there any way to rewrite and/or approximate those expressions with something that is not going to numerically diverge when $$x \approx 0$$?