# How can the relationship $\frac{\triangle G_t}{Y_t}=\triangle \left(\frac{G_t}{Y_t}\right)$ be established?

The calculation below is from a NBER paper.

If $$Y_t$$ grows at the constant rate $$g=r$$, then $$\triangle G_t+\frac{\triangle G_{t+1}}{1+r}+\frac{\triangle G_{t+2}}{(1+r)^2}+\dots$$ can be rewritten as $$Y_t \left(\triangle \frac{G_t}{Y_t}+\triangle \frac{G_{t+1}}{Y_{t+1}}+\triangle \frac{G_{t+2}}{Y_{t+2}}+\dots\right)$$ I think this calculation is from $$Y_{t+1}=Y_t (1+r)$$ and $$\frac{\triangle G_t}{Y_t}=\triangle \left(\frac{G_t}{Y_t}\right)$$.
How $$\frac{\triangle G_t}{Y_t}=\triangle \left(\frac{G_t}{Y_t}\right)$$ relationship can be established?
Or are there any other missing clues I didn't noticed?

How $$\frac{\triangle G_t}{Y_t}=\triangle \left(\frac{G_t}{Y_t}\right)$$ relationship can be established?
If $$\Delta G_t={G_t-G_{t-1}}$$ and $$Y_t$$ is fixed:
$$\frac{\Delta G_t}{Y_t}=\frac {G_t-G_{t-1}}{Y_t}=\frac {G_t}{Y_t}-\frac {G_{t-1}}{Y_t}=\Delta \left(\frac{G_t}{Y_t}\right)$$.