I came across this expression in Thomas' Calculus: early transcendentals. I understood the expressions describing the Brachistochrone curve up until this point where the derivation jumps from:
$\frac{ds}{dT}=\sqrt{2gy}$
Where T is time, ds is arc length differential and g is acceleration due to gravity to:
$dT=\frac{ds}{\sqrt{2gy}}=\frac{\sqrt{1+(dy/dx)^2}dx}{\sqrt{2gy}}$
The 2nd part of this is what confuses me as to how $ds$ becomes $\sqrt{1+(dy/dx)^2}dx$ in a single trivial step as I cannot think of a simple conversion from one to the other and neither is it explicitly described in the book.