Given d&gt;1, x ∈ I R d , and two multivariate functions u , v : I R d → I R d , what is the second derivative of f : I R d → I R f ( x ) = u ( x ) ⋅ v ( x ) = u ( x ) T v ( x )with respect to x?The first derivative is ∂ f ∂ x ( x ) = u ( x ) T ∂ v ∂ x ( x ) + v ( x ) T ∂ u ∂ x ( x )which is a row vector of size d when using the numerator layout.But I do not know how to differentiate again since ∂ u ∂ x and ∂ v ∂ x are 3-by-3 matrices...I tried to develop these equations in a simple situation where d=2, but I obtain a quite complicated matrix and I am looking for a clear formula, in any dimension d&gt;1, where only u,v and their derivatives appear.

Question

Given d&amp;gt;1,       x    ∈            I            R        d  , and two multivariate functions       u    ,      v    :            I            R        d    →            I            R        d  , what is the second derivative of   f  :            I            R        d    →      I        R    f  (      x    )  =      u    (      x    )  ⋅      v    (      x    )  =      u    (      x        )    T        v    (      x    )with respect to x?The first derivative is            ∂      f              ∂              x              (      x    )  =      u    (      x        )    T              ∂              v                    ∂              x              (      x    )  +      v    (      x        )    T              ∂              u                    ∂              x              (      x    )which is a row vector of size d when using the numerator layout.But I do not know how to differentiate again since             ∂              u                    ∂              x             and             ∂              v                    ∂              x             are 3-by-3 matrices...I tried to develop these equations in a simple situation where d=2, but I obtain a quite complicated matrix and I am looking for a clear formula, in any dimension d&amp;gt;1, where only u,v and their derivatives appear.

Luna Wells · Accepted Answer

It&#039;s easier to work in index notation. Write   f  =      ∑          i      =      1        d        u    i        v    i   so  (  ∇  f      )    j    =      ∑    i    [  (      ∂    j        u    i    )      v    i    +      u    i        ∂    j        v    i    ]  =  [  (  ∇  u  )  v  +  (  ∇  v  )  u      ]    j    ,i.e.   ∇  f  =  (  ∇  u  )  v  +  (  ∇  v  )  u. Now for the second derivative:  (      ∇    2    f      )          j      k        =      ∂    k    (  ∇  f      )    j    =      ∑    i    [  (      ∂    j        ∂    k        u    i    )      v    i    +      ∂    j        u    i        ∂    k        v    i    +      ∂    k        u    i        ∂    j        v    i    +      u    i        ∂    j        ∂    k        v    i    ]  .Note this includes rank-3 tensors we might denote       ∇    2    u  ,        ∇    2    v, but writing in terms of those without indices requires some clarification on the notation. For example, if we take   (      ∇    2    u      )          j      k      i        :=  (      ∂    j        ∂    k        u    i    ), we get      ∇    2    f  =  (      ∇    2    u  )  ⋅  v  +  (  ∇  u  )  ⋅  (  ∇  v  )  +  (  ∇  v  )  ⋅  (  ∇  u  )  +  (      ∇    2    v  )  ⋅  u  ,where each ⋅ contracts the factors&#039; last index. This is the general Leibniz rule with non-commuting factors.

Given d>1, x in RR^d, and two multivariate functions u,v: RR^d to RR^d, what is the second derivative of f:R^d to RR f(x)=u(x) * v(x)=u(x)^T v(x) with respect to x?

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