2nd derivative of function with 3 parametric variables?

Question

For the function $f(x,y,z)$, where $x,y,z$ are all functions of $t$, what is the 2nd derivative of $f$ with respect to $t$?

I know that the 1st derivative is $\frac {df}{dt} = \frac{\partial f}{\partial x} x'(t)+\frac{\partial f}{\partial y} y'(t)+\frac{\partial f}{\partial z} z'(t) $, But can't get my head around the how the chain rule would work for the 2nd derivative.

Answer 1

\begin{align*} \frac{\mathrm{d}}{\mathrm{d}t} f(x(t),y(t),z(t)) &= \frac{\partial f}{\partial x} \frac{\mathrm{d}x}{\mathrm{d}t} + \frac{\partial f}{\partial y} \frac{\mathrm{d}y}{\mathrm{d}t} + \frac{\partial f}{\partial z} \frac{\mathrm{d}z}{\mathrm{d}t} \text{.} \\ \frac{\mathrm{d}^2}{\mathrm{d}t^2} f(x(t),y(t),z(t)) &= \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial f}{\partial x} \frac{\mathrm{d}x}{\mathrm{d}t} + \frac{\partial f}{\partial y} \frac{\mathrm{d}y}{\mathrm{d}t} + \frac{\partial f}{\partial z} \frac{\mathrm{d}z}{\mathrm{d}t} \right) \\ &= \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial f}{\partial x} \frac{\mathrm{d}x}{\mathrm{d}t} \right) + \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial f}{\partial y} \frac{\mathrm{d}y}{\mathrm{d}t}\right) + \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial f}{\partial z} \frac{\mathrm{d}z}{\mathrm{d}t} \right) \text{.} \\ \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial f}{\partial x} \frac{\mathrm{d}x}{\mathrm{d}t} \right) &= \frac{\partial f}{\partial x} \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\mathrm{d}x}{\mathrm{d}t} \right) + \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial f}{\partial x} \right) \frac{\mathrm{d}x}{\mathrm{d}t} \\ &= \frac{\partial f}{\partial x} \frac{\mathrm{d}^2x}{\mathrm{d}t^2} + \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial f}{\partial x} \right) \frac{\mathrm{d}x}{\mathrm{d}t} \text{.} \\ \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial f}{\partial x} \right) &= \left( \frac{\partial \frac{\partial f}{\partial x}}{\partial x} \frac{\mathrm{d}x}{\mathrm{d}t} + \frac{\partial \frac{\partial f}{\partial x}}{\partial y} \frac{\mathrm{d}y}{\mathrm{d}t} + \frac{\partial \frac{\partial f}{\partial x}}{\partial z} \frac{\mathrm{d}z}{\mathrm{d}t} \right) \\ &= \left( \frac{\partial^2 f}{\partial x^2} \frac{\mathrm{d}x}{\mathrm{d}t} + \frac{\partial^2 f}{\partial x \partial y} \frac{\mathrm{d}y}{\mathrm{d}t} + \frac{\partial^2 f }{\partial x \partial z} \frac{\mathrm{d}z}{\mathrm{d}t} \right) \text{, so} \\ \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial f}{\partial x} \frac{\mathrm{d}x}{\mathrm{d}t} \right) &= \frac{\partial f}{\partial x} \frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + \left( \frac{\partial^2 f}{\partial x^2} \frac{\mathrm{d}x}{\mathrm{d}t} + \frac{\partial^2 f}{\partial x \partial y} \frac{\mathrm{d}y}{\mathrm{d}t} + \frac{\partial^2 f }{\partial x \partial z} \frac{\mathrm{d}z}{\mathrm{d}t} \right) \frac{\mathrm{d}x}{\mathrm{d}t} \text{, so} \\ \frac{\mathrm{d}^2}{\mathrm{d}t^2} f(x(t),y(t),z(t)) &= \frac{\partial f}{\partial x} \frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + \left( \frac{\partial^2 f}{\partial x^2} \frac{\mathrm{d}x}{\mathrm{d}t} + \frac{\partial^2 f}{\partial x \partial y} \frac{\mathrm{d}y}{\mathrm{d}t} + \frac{\partial^2 f }{\partial x \partial z} \frac{\mathrm{d}z}{\mathrm{d}t} \right) \frac{\mathrm{d}x}{\mathrm{d}t} + \frac{\partial f}{\partial y} \frac{\mathrm{d}^2 y}{\mathrm{d}t^2} + \left( \frac{\partial^2 f}{\partial x \partial y} \frac{\mathrm{d}x}{\mathrm{d}t} + \frac{\partial^2 f}{\partial y^2} \frac{\mathrm{d}y}{\mathrm{d}t} + \frac{\partial^2 f }{\partial y \partial z} \frac{\mathrm{d}z}{\mathrm{d}t} \right) \frac{\mathrm{d}y}{\mathrm{d}t} + \frac{\partial f}{\partial z} \frac{\mathrm{d}^2 z}{\mathrm{d}t^2} + \left( \frac{\partial^2 f}{\partial x \partial z} \frac{\mathrm{d}x}{\mathrm{d}t} + \frac{\partial^2 f}{\partial y \partial z} \frac{\mathrm{d}y}{\mathrm{d}t} + \frac{\partial^2 f }{\partial z^2} \frac{\mathrm{d}z}{\mathrm{d}t} \right) \frac{\mathrm{d}z}{\mathrm{d}t} \text{.} \end{align*}