\begin{align}
f(x,y) &= (x - 1)ye^{-\frac{1}{2}[(x-1)^2+y^2]} = xye^{-\frac{1}{2}[(x-1)^2+y^2]} - ye^{-\frac{1}{2}[(x-1)^2+y^2]} , \\
f_x(x,y) &= ye^{-\frac{1}{2}[(x-1)^2+y^2]} - xy(x-1)e^{-\frac{1}{2}[(x-1)^2+y^2]} + (x-1)ye^{-\frac{1}{2}[(x-1)^2+y^2]} \\
&= y[1 - x(x-1) + (x-1)]e^{-\frac{1}{2}[(x-1)^2+y^2]} \\
&= y[- x^2 + 2x]e^{-\frac{1}{2}[(x-1)^2+y^2]} \\
&= -xy[x - 2]e^{-\frac{1}{2}[(x-1)^2+y^2]} \\
f_y(x,y) &= (x - 1)e^{-\frac{1}{2}[(x-1)^2+y^2]} - (x - 1)y^2e^{-\frac{1}{2}[(x-1)^2+y^2]} \\

@HsMjstyMstdn \begin{align}
\theta(t) &= \theta_0 \cos(\omega_0 t) + \frac{\Omega_0}{\omega_0} \sin(\omega_0 t) \\
&= [A \cos(\delta)] \cos(\omega_0 t) + [A \sin(\delta) ]\sin(\omega_0 t) \\
\theta_0 &= A \cos(\delta) , \\
\frac{\Omega_0}{\omega_0} &= A \sin(\delta) , \\
\theta_0^2+(\frac{\Omega_0}{\omega_0})^2 &= A^2 \cos^2(\delta) + A^2\sin^2(\delta) = A^2 \to A = \sqrt{\theta_0^2+(\frac{\Omega_0}{\omega_0})^2}, \\
\frac{A \sin(\delta)}{A \cos(\delta)} &= \tan(\delta) = \frac{\frac{\Omega_0}{\omega_0} }{\theta_0} = \frac{\Omega_0}{\omega_0 \theta_0} \to \delta = \tan^{-1} ( \frac{\Omega_0…