Polar coordinates, Differentiation

Question

Can someone clarify this step for me please,

"The polar coordinate r satisfies $r^2=x^2+y^2$, so by differentiating with respect to t we get $r\cdot\dot r=x\cdot\dot x+y\cdot\dot y$"

I am totally lost...and I realize this should be simple, I'm just missing something..

Answer 1

Use implicit differentiation $$r^2(t) = x^2(t) + y^2(t) \Rightarrow 2r(t)r' = 2x(t)x' + 2y(t)y' \Rightarrow r r' = x x'+yy'$$

Answer 2

Write it explicitly: $\frac{d}{dt}(r(t)^2) = 2 \times r(t) \times \frac{dr}{dt}$ Same treatment for the other two.