Question

What is the second derivative of a parametric function of 2 variables?

Answer 1

If we have some differentiable function f(x, y) dependent upon some differential functions x(t) and y(t), it's first derivative with respect to t is:\[\frac{ dz }{ dt } = \frac{ \partial z }{ \partial x }\frac{ dx }{ dt } + \frac{ \partial z }{ \partial y }\frac{ dy }{ dt }\]This is an obvious application of the chain rule. However, the function's second derivative is not so obvious.\[\frac{ d^2 z }{ dt^2 } = \frac{ \partial^2 f }{ \partial x^2 } \left( \frac{ dx }{ dt } \right)^2+ \frac{ \partial^2 f }{ \partial y^2 } \left( \frac{ dy }{ dt } \right)^2 + \left( \frac{\partial^2 f }{ \partial y \partial x } + \frac{ \partial^2 f }{ \partial x \partial y } \right) \frac{ dx }{ dt }\frac{ dy }{ dt } + \frac{ \partial f }{ \partial x } \frac{ d^2 x }{ dt^2 } + \frac{ \partial f }{ \partial y } \frac{ d^2 y }{ dt^2 }\]I can see where the first and last 2 terms of the second derivative come from, but where does the middle term come from?

Answer 2

The last term, which isn't visible on my previous post, is: \[\frac{ \partial f }{ \partial y } \frac{ d^2 y }{ dt^2 }\]