I need to show that
$$\dfrac{\Gamma\left(\alpha_1 + \alpha_2\right)}{\Gamma\left(\alpha_1\right)\Gamma\left(\alpha_2\right)}\left[\dfrac{\tau y^{\tau \alpha_1}\delta^{\tau \alpha_2}}{y\left(y^{\tau} + \delta^{\tau}\right)^{\alpha_1 + \alpha_2}}\right] = \dfrac{\Gamma\left(\alpha_1 + \alpha_2\right)}{\Gamma\left(\alpha_1\right)\Gamma\left(\alpha_2\right)}\left\{\dfrac{\tau \left(y/\delta\right)^{\tau \alpha_1}}{y\left[1+\left(y/\delta\right)^{\tau}\right]^{\alpha_1 + \alpha_2}}\right\}\text{.}$$ I don't see what is necessary to get the expression on the right. (I've been studying for this exam too long.) Looks like some factoring, but it looks very messy.
Use that $$(y^{\tau}+\delta^{\tau})^{\alpha_{1}+\alpha_{2}}=\delta^{\tau(\alpha_{1}+\alpha_{2})}\left(1+\left(\frac{y}{\delta}\right)^{\tau}\right)^{\alpha_{1}+\alpha_{2}}$$
and $$\frac{y^{\tau\alpha_{1}}\delta^{\tau\alpha_{2}}}{\delta^{\tau(\alpha_{1}+\alpha_{2})}}=\left(\frac{y}{\delta}\right)^{\tau\alpha_{1}}.$$