Question

If r = -2cost, then dy/dx=

Answer 1

Hello watson,

You can use the transformations from rectilinear to polar coordinates:\[x=r \cos t, y = r \sin t\]Given this, it follows\[x^2+y^2=r^2\cos^ t + r^2 \sin^2 t= r^2 (\cos^2 t + \sin^2 t) = r^2\]Your original equation can be rewritten as,\[r=-2\frac{x}{r} \rightarrow r^2=-2x \rightarrow x^2+y^2=-2x \rightarrow y^2=-2x-x^2\]Operating on the left and right sides with the derivative with respect to x,\[\frac{d}{dx}y^2=\frac{d}{dx}(-2x-x^2)\rightarrow \frac{dy^2}{dy}\frac{dy}{dx}=-2-2x\]\[\rightarrow 2y \frac{dy}{dx}=-2(1+x) \rightarrow \frac{dy}{dx}=-\frac{1+x}{y}\]