linear approximation for ratio $\frac{(1+x)^n}{(1+y)^m}$

Question

I have seen people use the approximation $\frac{(1+x)^n}{(1+y)^m}\approx1+nx-my$ where x and y are close to 0. I know $(1+x)^n \approx 1+nx$, but not sure about the ratio to difference approximation. Can someone please give an derivation?

Answer 1

For a formal proof, it results from Taylor's formula at order $1$ for two variables:

Set $f(x,y)=\dfrac{(1+x)^n}{(1+y)^m}.\;$ Taylor's formula at order $1$ asserts that $$f(x,y)=f(0,0)+\frac{\partial f}{\partial x}(0,0)\,x+\frac{\partial f}{\partial x}(0,0)\,y+o\bigl(\|(x,y)\|\bigr).$$ Here, we have $$\begin{cases}\frac{\partial f}{\partial x}=\frac{n(1+x)^{n-1}}{(1+y)^m},&\text{whence } \;\frac{\partial f}{\partial x}(0,0)=n, \\[1.5ex] \frac{\partial f}{\partial y}=-\frac{m(1+x)^n}{(1+y)^{m-1}},&\text{whence } \;\frac{\partial f}{\partial y}(0,0)= -m, \end{cases}$$ so that Taylor's formula becomes $$f(x,y)=1+nx-my+o\bigl(\|(x,y)\|\bigr).$$