I have the following function
$$f(x_1,x_2)=\frac{ax_1}{ax_1+bx_2}$$
Where the partial derivative of $f$ with respect to $x_2$ is
$$\frac{\partial f}{\partial x_2}=-\frac{abx_1}{(ax_1+bx_2)^2}$$
But why is the absolute value of $\frac{\partial f}{\partial x_2}$ (i.e. the size of the effect of increasing $x_2$) increasing in $a$?
It seems counter-intuitive since clearly as $a \rightarrow \infty$, the effect on $f$ increasing $x_2$ will tend to $0$. What am I overlooking?
You have $a$ in both numerator and denominator. And, since its power in numerator is equal to $1$ and in denominator this power is equal to $2$, you can see that, indeed, the partial derivative $\partial_{x_2} f$ vanishes as $|a|\to\infty$.