implicit differentiation $\large\frac{-x}{y}\frac{dx}{dt} = \frac{dy}{dt}$ to $\small(\cos{\theta})\large\frac{d\theta}{dt} = \frac{dy/dt}{13}$

Question

Suppose $\frac{y}{13} = \sin{\theta}$.

Please show the steps to implicitly differentiate $\large\frac{-x}{y}\frac{dx}{dt} = \frac{dy}{dt}$ with respect to $t$ to reach $\small(\cos{\theta})\large\frac{d\theta}{dt} = \frac{dy/dt}{13}$.

Starting with $\frac{-x}{13 \sin{\theta}} \frac{dx}{dt} = \frac{dy}{dt}$, I am not sure how to proceed next.

Answer 1

Ignore the $\large\frac{-x}{y}\frac{dx}{dt} = \frac{dy}{dt}$ equation and focus on: $$ \frac{y}{13} = \sin{\theta} $$

Implicitly differentiating each side with respect to $t$, we directly obtain: $$ \frac{dy/dt}{13} = \cos\theta \frac{d\theta}{dt} $$