We have the following function:
$$f(\boldsymbol{x}) = \left(\frac{x_1}{x_2}\right)^\sigma + x_1$$
We would like to derivate it by the fraction of $x_1$ and $x_2$:
$$\frac{\partial \left[ \left( \frac{x_1}{x_2} \right)^\sigma + x_1 \right]}{\partial \frac{x_1}{x_2}}$$
The derivation of the first term seems easy to accomplish:
$$\frac{\partial \left( \frac{x_1}{x_2} \right)^\sigma}{\partial \frac{x_1}{x_2}} = \sigma \left( \frac{x_1}{x_2} \right)^{\sigma-1}$$
But what would be the following equal to:
$$\frac{\partial x_1}{\partial \frac{x_1}{x_2}} = \boldsymbol{?}$$
HINT: Let $z=\frac{x_1}{x_2}$. You can then define $x_1(z)=zx_2$ and differentiate $f(z,x_1(z))$ with respect to $z$, right?