Question

if x=a cos^3 t and y= a sin^3 t, where a is a constant, find dy/dx & d^2y/dx^2 in term of t

Answer 1

Question: if x=a cos^3 t and y= a sin^3 t, where a is a constant, find dy/dx & d^2y/dx^2 in term of t.

x = a.cos^3(t)

x' = -3a.sin(t).cos^2(t)

y = a.sin^3(t)

y' = 3a.cos(t).sin^2(t)

dy/dx = y'/x' = {3a.cos(t).sin^2(t)}/{-3a.sin(t).cos^2(t)} = -tan(t)

dy/dx = -tan(t)

d^2y/dx^2 = d(dy/dx)/dx = d(dy/dx)/dt.dt/dx = sec^2(t)/(-3a.sin(t).cos^2(t))

d^2y/dx^2 = (-1)/(3a*cos^4(t).sin(t)