Using the chain rule to find the second derivative of a parametric multivariable function.

[burt]

I was given the following:
Use the chain rule twice to find \(g"(t)\) of \(g(t)=f(x(t),y(t),z(t))\).

This is my work: \(g'(t)=F_x\frac{dx}{dt}+F_y\frac{dy}{dt}+F_z\frac{dz}{dt}\) \[g"(t)=F_{xx}\frac{dx}{dt}+F_x\frac{d^2x}{dt^2}+F_{yy}\frac{dy}{dt}+F_y\frac{d^2y}{dt^2}+F_{zz}\frac{dz}{dt}+F_z\frac{d^2z}{dt^2}\]

Is this the correct way to do this? Am I misunderstanding how to use the chain rule on a multivariable function?

 

[Romsek]

All looks good

One little thing you might note, notation wise, is that it's common to use

\(\displaystyle \dot{x} = \dfrac{dx}{dt}\\
\ddot{x} = \dfrac{d^2x}{dt^2}\)

 

[burt]

All looks good

One little thing you might note, notation wise, is that it's common to use

\(\displaystyle \dot{x} = \dfrac{dx}{dt}\\
\ddot{x} = \dfrac{d^2x}{dt^2}\)

Cool. I didn't know that!