If x = a(cos t + t sin t) and y = a(sin t - t cos t), 0 < t < π/2, find d²x/dt², d²y/dt² and d²y/dx².

x = a(cos t + t sin t), y = a(sin t - t cos t)
dx/dt = a(-sin t + t cos t + sin t) = at cos t
d²x/dt² = d(dx/dt)/dt = a(cos t - t sin t) = a(-t sin t + cos t)
dy/dt = a(cos t - cos t + t sin t) = at sin t
d²y/dt² = a(sin t + t cos t)
d²y/dx² = d(dy/dx)/dx = d(dy/dt * dt/dx)/dx = d(at sin t / at cos t)/dt * dt/dx = d(tan t)/dt * 1/(at cos t) = sec²t / (at cos t) = sec³t / at