Question

A point is sliding along a curve y^2 = 1 + cos^3(2x) at whatever speed (even depending on time). Compute the ratio of the rates of change of the vertical and horizontal coordinate with respect to time when the point is at (x, y).

Answer 1

"the ratio of the rates of change of the vertical and horizontal coordinate with respect to time"
means
find [dy/dt] / [dx/dt]

so for y^2 = 1 + (cos2x)^3
2y dy/dt = 3(cos2x)^2 (-sin2x)(2) dx/dt

[dy/dt] / [dx/dt] = -6(sin2x)(cos2x)^2 /(2y)
= -3(sin2x)(cos2x)^2 /y