Rewriting a Derivative with Logs & Sums & a Scalar

Question

Consider the following derivative that contains two random variables ($x$ and $y$) and one constant scalar ($\beta$):

$\frac{\partial}{\partial \log(y)} \, \log(x+y)\beta$.

Let us define this derivative as $\Delta$, so:

$\Delta \equiv \frac{\partial}{\partial \log(y)} \, \log(x+y)\beta$.

Question: Can we rewrite the above definition to solve for the constant $\beta$?

In other words, how do we define $\beta$ as a function of $\Delta, x,$ and $y$?

Answer 1

Hint : \begin{align*} \log(x+y) &= \log\left(x+e^{\log(y)}\right) \end{align*} Using this you get \begin{align*} \beta = \Delta \frac{x+y}{y} \end{align*}