Final answer:
In mathematics, finding the second derivative of a parametric function such as x(t)=sin(9t) and y(t)=2cos(9t) requires first determining the first and second derivatives concerning t, and then applying the formula (d^2y/dx^2) = (d^2y/dt^2) / (dx/dt)^3.
Step-by-step explanation:
The subject of the question is Mathematics, specifically finding the second derivative of a parametric function without eliminating the parameter. Given x(t)=sin(9t) and y(t)=2cos(9t), we first find the first derivatives concerning t, dx/dt, and dy/dt, and then the second derivative of y concerning t, d^2y/dt^2. Since d^2y/dx^2 = (d^2y/dt^2) / (dx/dt)^3 provided that dx/dt is not zero, we can substitute our earlier derivatives into this formula to find the answer.
- First derivatives: dx/dt = 9cos(9t), dy/dt = -18sin(9t)
- Second derivative of y with respect to t: d^2y/dt^2 = -162cos(9t)
- Using the formula d^2y/dx^2 = (d^2y/dt^2) / (dx/dt)^3
Substituting values in, we can calculate (d^2y/dx^2).