Can you explain the derivation of the second derivative of a parametric equation?

Answer 1

The second derivative of a parametric equation, such as x = f(t) and y = f(t), goes as follows:

d²y/dx² = (d/dt(dy/dx))/(dx/dt)

That may look a bit confusing, especially with this site's limited functionality, but allow me to elaborate:

The second derivative of a parametric function is equal the derivative with respect to t of the first derivative divided by the derivative of x with respect to t.

Let's do an example to make it a bit clearer.

Take x = t² and y = t^3:

dx/dt = 2t, dy/dt = 3t², and thus dy/dx = 3t²/2t = 3t/2.

d²y/dx² = (d/dt(3t/2))/(dx/dt) = (3/2)/(2t) = 3/4t.

Hopefully this makes things a bit clearer. Feel free to follow up if this is still confusing or I did a poor job explaining.