A function is defined parametrically as x = 4 sin(3t), y = 2 cos(3t). Find and simplify d^2 y/dx^2 in terms of t and y.

We first need to find dy/dx and we use the fact that dy/dx = dy/dt * dt/dx. So we have dy/dt = -6sin(3t) and dx/dt = 12cos(3t). Substituing these in we have dy/dx = -6*sin(3t)1/(12cos(3t) which simplifies to -(1/2)*tan(3t).
We now have dy/dx = -tan(3t)/2.

To find the second derivative we again have to derive w.r.t. t.
d²y/dx² = d/dt(dy/dx)dt/dx and we have already found dt/dx as 1/(12cos(3t)).
So we have d/dt(-tan(3t)/2) = -3sec² (3t)/2 and thus
d²y/ dx² = -3sec² (3t)/2*(12*cos(3t)) which simplifies to - sec³ (3t) / 8 in terms of t.
In terms of y, we know y = 2 cos(3t) and therefore cos(3t) = y/2, thus cos³(3t) = y³/8 and sec³(3t) = 8/y³
Finally we have d²y/dx²=-1/y³.