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Second partials chain rule

  1. Nov 2, 2006 #1
    This is a stupid question but...

    The regular multivariable chain rule is:

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

    Now, are there formulae for the second partials [tex]u_{xx}, u_{xy}, u_{yy}[/tex]

    I just want to check myself (this isn't a homework problem, though it is study for a class) Thanks
    Last edited: Nov 2, 2006
  2. jcsd
  3. Nov 2, 2006 #2


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    Easy, just use product rule:

    [tex]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}[/tex]
  4. Nov 2, 2006 #3


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    If I recall correctly though, [tex]u_{vx}[/tex] is really [tex]u_{vv}v_x + u_{vw}w_x[/tex]
  5. Nov 2, 2006 #4


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    Yeah, it probably is.
  6. Nov 2, 2006 #5
    Hmm, yeah. I knew it was straightforward, but it seemed too simple for some reason.
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