This is a problem from my practice midterm. No answers are given. Can someone please tell me if I have answered this problem correctly?

S=Distance between P & Airplane

$\displaystyle

\displaystyle sin \theta=\frac{5000}{S}

$

I take the derivative of $\displaystyle \theta$ & S with respect to $\displaystyle \theta$

$\displaystyle

\displaystyle cos \theta=\frac{5000}{\frac{dS}{d\theta}}

$

$\displaystyle

\displaystyle \frac{dS}{d\theta}=\frac{5000}{cos\theta}=5773.5

$

Is my work correct?

Is it OK that I only took the derivative of S (with respect to theta) on the denominator only on the right hand side, or should I have taken the derivative of the entire right hand side using the quotient rule?

In my practice problems, when doing implicit derivatives, we take the derivatives of both sides in their entirety. However, I am watching a youtube video on related rates regarding the volume of a balloon, and the guy only takes the derivatives of the two variables he is interested in. Can someone please elaborate on this?

Thank you!