ds=√(dx)^2+(dy)^2
=√(drcosθ)^2+(drsinθ)^2
=√(rdcosθ+cosθdr)^2+(rdsinθ+sinθdr)^2
=√(-rsinθdθ+r'cosθdθ)^2+(rcosθdθ+r'sinθdθ)^2
=(√r^2+r'^2)dθ
Method 2:
The formula ds = (√ x' (t) 2+y' (t) 2) dt.
Where x(θ)=r(θ)cos(θ) and y(θ)=r(θ)sin(θ).
Direct substitution can get ds = (√ r 2+r' 2) d θ.
The two methods are actually equivalent, and there is not much difference.