# Second partials chain rule

This is a stupid question but...

The regular multivariable chain rule is:

$$u_x = u_v v_x + u_w w_x$$ and $$u_y = u_v v_y + u_w w_y$$ where $$u(v(x, y), w(x, y))$$

Now, are there formulae for the second partials $$u_{xx}, u_{xy}, u_{yy}$$

I just want to check myself (this isn't a homework problem, though it is study for a class) Thanks

Last edited:

Related Calculus and Beyond Homework Help News on Phys.org
AKG
Homework Helper
Easy, just use product rule:

$$u_{xx} = (u_x)_x = (u_vv_x + u_ww_x)_x = u_{vx}v_x + u_vv_{xx} + u_{wx}w_x + u_ww_{xx}$$

Office_Shredder
Staff Emeritus
Gold Member
AKG said:
Easy, just use product rule:

$$u_{xx} = (u_x)_x = (u_vv_x + u_ww_x)_x = u_{vx}v_x + u_vv_{xx} + u_{wx}w_x + u_ww_{xx}$$
If I recall correctly though, $$u_{vx}$$ is really $$u_{vv}v_x + u_{vw}w_x$$

AKG