\begin{align*}
    \frac{\partial}{\partial r_1} r_{12}
    &= \frac{\partial}{\partial r_1} \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2} \\
    &= \frac{\partial}{\partial r_1} \sqrt{(r_1\sin\theta_1\cos\phi_1 - x_2)^2 + (r_1\sin\theta_1\sin\phi_1 - y_2)^2 + (r_1\cos\theta_1 - z_2)^2} \\
    &= \left(\frac{1}{2r_{12}}\right) \frac{\partial}{\partial r_1} \Big[ (r_1\sin\theta_1\cos\phi_1 - x_2)^2 + (r_1\sin\theta_1\sin\phi_1 - y_2)^2 + (r_1\cos\theta_1 - z_2)^2 \Big] \\
    &= \frac{1}{2r_{12}} \Big[ 2(x_1 - x_2)(\sin\theta_1\cos\phi_1) + 2(y_1 - y_2)(\sin\theta_1\sin\phi_1) + 2(z_1 - z_2)(\cos\theta_1) \Big] \\
    &= \frac{1}{r_{12}} \left[ (x_1 - x_2)\frac{x_1}{r_1} + (y_1 - y_2)\frac{y_1}{r_1} + (z_1 - z_2)\frac{z_1}{r_1} \right] \\
    &= \frac{1}{r_1 r_{12}} \Big[ r_1^2 - x_1x_2 - y_1y_2 - z_1z_2 \Big] \\
    &= \frac{1}{r_1 r_{12}} \Big[ r_1^2 - \vec r_1 \vec r_2 \Big] \\
\end{align*}