Multivariable Chain Rule

[TranscendArcu]

Suppose I have

F(x,y) and y=y(t) and x=x(t)

Therefore,

F_t = F_x*x_t + F_y*y_t. Right?

Can I write

F_tt = (F_xx*x_t + F_yy*y_t)*x_t + F_x*x_tt + (F_xx*x_t + F_yy*y_t)*y_t + F_y*y_tt

?

Basically I'm trying to figure out the second derivative by chain rule.

 

[Dick]

Shouldn't there be some F_xy's and F_yx's in there?

 

[TranscendArcu]

You're right. I meant to write:


F_tt = (F_xx*x_t + F_xy*y_t)*x_t + F_x*x_tt + (F_yx*x_t + F_yy*y_t)*y_t + F_y*y_tt

 

[Dick]

You're right. I meant to write:


F_tt = (F_xx*x_t + F_xy*y_t)*x_t + F_x*x_tt + (F_yx*x_t + F_yy*y_t)*y_t + F_y*y_tt

Looks ok now, good job. Just product rule and chain rule, yes?