Question

x = rcos (theta) and y = rsin (theta), find dy/dx

Answer 1

Observe that \[\begin{align*}x^2+y^2&=r^2\cos^2\theta+r^2\sin^2\theta\\&=r^2(\cos^2\theta+\sin^2\theta)\\&=r^2\end{align*}\]
Now, you can do some implicit differentiation; differentiate both sides with respect to \(x\). Keep in mind that \(r\) is a constant:
\[\frac{d}{dx}\left[x^2\right]+\frac{d}{dx}\left[y^2\right]=\frac{d}{dx}\left[r^2\right]\]