To give some intuition on the Fa\`a di Bruno formula, we illustrate \cref{eq:taylor-mode-composition} for higher orders here:
\vspace{2ex}
\begin{tiny}
  \begin{align*}
    \vx_{0}
    &\to&
          \vh_{0}
          =
          \vh(\vx_{0})
    &\to&
          \vg_{0}
          =
          \vg(\vh_{0})
          =
    &\vf_{0}
      =
      \vf(\vx_{0})
    \\
    \vx_{1}
    &\to&
          \vh_{1}
          =
          \langle \partial \vh, \vx_{1} \rangle
    &\to&
          \vg_{1}
          =
          \langle \partial \vg, \vh_{1} \rangle
          =
    &\vf_{1}
      =
      \langle \partial \vf, \vx_{1} \rangle
    \\
    \vx_{2}
    &\to&
          \vh_{2}
          =
          \begin{matrix}
            \langle \partial^{2} \vh, \vx_{1}^{\otimes2} \rangle
            \\
            +
            \langle \partial \vh, \vx_{2} \rangle
          \end{matrix}
    &\to&
          \vg_{2}
          =
          \begin{matrix}
            \langle \partial^{2} \vg, \vh_{1}^{\otimes2} \rangle
            \\
            +
            \langle \partial \vg, \vh_{2} \rangle
          \end{matrix}
          =
    &\vf_{2}
      =
      \begin{matrix}
        \langle \partial^{2} \vf, \vx_{1}^{\otimes2} \rangle
        \\
        +
        \langle \partial \vf, \vx_{2} \rangle
      \end{matrix}
    \\
    \vx_{3}
    &\to&
          \vh_{3}
          =
          \begin{matrix}
            \langle \partial^{3} \vh, \vx_{1}^{\otimes3} \rangle
            \\
            +
            3 \langle \partial^{2} \vh, \vx_{1}\otimes \vx_{2} \rangle
            \\
            +
            \langle \partial \vh, \vx_{3} \rangle
          \end{matrix}
    &\to&
          \vg_{3}
          =
          \begin{matrix}
            \langle \partial^{3} \vg, \vh_{1}^{\otimes3} \rangle
            \\
            +
            3 \langle \partial^{2} \vg, \vh_{1}\otimes \vh_{2} \rangle
            \\
            +
            \langle \partial \vg, \vh_{3} \rangle
          \end{matrix}
          =
    &\vf_{3}
      =
      \begin{matrix}
        \langle \partial^{3} \vf, \vx_{1}^{\otimes3} \rangle
        \\
        +
        3 \langle \partial^{2} \vf, \vx_{1}\otimes \vx_{2} \rangle
        \\
        +
        \langle \partial \vf, \vx_{3} \rangle
      \end{matrix}
    \\
    \vx_{4}
    &\to&
          \vh_{4}
          =
          \begin{matrix}
            \langle \partial^{4} \vh, \vx_{1}^{\otimes4} \rangle
            \\
            +
            6 \langle \partial^{3} \vh, \vx_{1}^{\otimes2}\otimes \vx_{2} \rangle
            \\
            +
            4 \langle \partial^{2} \vh, \vx_{1}\otimes \vx_{3} \rangle
            \\
            +
            3 \langle \partial^{2} \vh, \vx_{2}^{\otimes2} \rangle
            \\
            +
            \langle \partial \vh, \vx_{4} \rangle
          \end{matrix}
    &\to&
          \vg_{4}
          =
          \begin{matrix}
            \langle \partial^{4} \vg, \vh_{1}^{\otimes4} \rangle
            \\
            +
            6 \langle \partial^{3} \vg, \vh_{1}^{\otimes2}\otimes \vh_{2} \rangle
            \\
            +
            4 \langle \partial^{2} \vg, \vh_{1}\otimes \vh_{3} \rangle
            \\
            +
            3 \langle \partial^{2} \vg, \vh_{2}^{\otimes2} \rangle
            \\
            +
            \langle \partial \vg, \vh_{4} \rangle
          \end{matrix}
          =
    &\vf_{4}
      =
      \begin{matrix}
        \langle \partial^{4} \vf, \vx_{1}^{\otimes4} \rangle
        \\
        +
        6 \langle \partial^{3} \vf, \vx_{1}^{\otimes2}\otimes \vx_{2} \rangle
        \\
        +
        4 \langle \partial^{2} \vf, \vx_{1}\otimes \vx_{3} \rangle
        \\
        +
        3 \langle \partial^{2} \vf, \vx_{2}^{\otimes2} \rangle
        \\
        +
        \langle \partial \vf, \vx_{4} \rangle
      \end{matrix}
    \\
    \vx_{5}
    &\to&
          \vh_{5}
          =
          \begin{matrix}
            \langle \partial^{5} \vh, \vx_{1}^{\otimes5} \rangle
            \\
            +
            10 \langle \partial^{4} \vh, \vx_{1}^{\otimes3}\otimes \vx_{2} \rangle
            \\
            +
            10 \langle \partial^{3} \vh, \vx_{1}^{\otimes2}\otimes \vx_{3} \rangle
            \\
            +
            15 \langle \partial^{3} \vh, \vx_{1}\otimes \vx_{2}^{\otimes2} \rangle
            \\
            +
            5 \langle \partial^{2} \vh, \vx_{1}\otimes \vx_{4} \rangle
            \\
            +
            10 \langle \partial^{2} \vh, \vx_{2}\otimes \vx_{3} \rangle
            \\
            +
            \langle \partial \vh, \vx_{5} \rangle
          \end{matrix}
    &\to&
          \vg_{5}
          =
          \begin{matrix}
            \langle \partial^{5} \vg, \vh_{1}^{\otimes5} \rangle
            \\
            +
            10 \langle \partial^{4} \vg, \vh_{1}^{\otimes3}\otimes \vh_{2} \rangle
            \\
            +
            10 \langle \partial^{3} \vg, \vh_{1}^{\otimes2}\otimes \vh_{3} \rangle
            \\
            +
            15 \langle \partial^{3} \vg, \vh_{1}\otimes \vh_{2}^{\otimes2} \rangle
            \\
            +
            5 \langle \partial^{2} \vg, \vh_{1}\otimes \vh_{4} \rangle
            \\
            +
            10 \langle \partial^{2} \vg, \vh_{2}\otimes \vh_{3} \rangle
            \\
            +
            \langle \partial \vg, \vh_{5} \rangle
          \end{matrix}
          =
    &\vf_{5}
      =
      \begin{matrix}
        \langle \partial^{5} \vf, \vx_{1}^{\otimes5} \rangle
        \\
        +
        10 \langle \partial^{4} \vf, \vx_{1}^{\otimes3}\otimes \vx_{2} \rangle
        \\
        +
        10 \langle \partial^{3} \vf, \vx_{1}^{\otimes2}\otimes \vx_{3} \rangle
        \\
        +
        15 \langle \partial^{3} \vf, \vx_{1}\otimes \vx_{2}^{\otimes2} \rangle
        \\
        +
        5 \langle \partial^{2} \vf, \vx_{1}\otimes \vx_{4} \rangle
        \\
        +
        10 \langle \partial^{2} \vf, \vx_{2}\otimes \vx_{3} \rangle
        \\
        +
        \langle \partial \vf, \vx_{5} \rangle
      \end{matrix}
    \\
    \vx_{6}
    &\to&
          \vh_{6}
          =
          \begin{matrix}
            \langle \partial^{6} \vh, \vx_{1}^{\otimes6} \rangle
            \\
            +
            15 \langle \partial^{5} \vh, \vx_{1}^{\otimes4}\otimes \vx_{2} \rangle
            \\
            +
            20 \langle \partial^{4} \vh, \vx_{1}^{\otimes3}\otimes \vx_{3} \rangle
            \\
            +
            45 \langle \partial^{4} \vh, \vx_{1}^{\otimes2}\otimes \vx_{2}^{\otimes2} \rangle
            \\
            +
            15 \langle \partial^{3} \vh, \vx_{1}^{\otimes2}\otimes \vx_{4} \rangle
            \\
            +
            60 \langle \partial^{3} \vh, \vx_{1}\otimes \vx_{2}\otimes \vx_{3} \rangle
            \\
            +
            15 \langle \partial^{3} \vh, \vx_{2}^{\otimes3} \rangle
            \\
            +
            6 \langle \partial^{2} \vh, \vx_{1}\otimes \vx_{5} \rangle
            \\
            +
            15 \langle \partial^{2} \vh, \vx_{2}\otimes \vx_{4} \rangle
            \\
            +
            10 \langle \partial^{2} \vh, \vx_{3}^{\otimes2} \rangle
            \\
            +
            \langle \partial \vh, \vx_{6} \rangle
          \end{matrix}
    &\to&
          \vg_{6}
          =
          \begin{matrix}
            \langle \partial^{6} \vg, \vh_{1}^{\otimes6} \rangle
            \\
            +
            15 \langle \partial^{5} \vg, \vh_{1}^{\otimes4}\otimes \vh_{2} \rangle
            \\
            +
            20 \langle \partial^{4} \vg, \vh_{1}^{\otimes3}\otimes \vh_{3} \rangle
            \\
            +
            45 \langle \partial^{4} \vg, \vh_{1}^{\otimes2}\otimes \vh_{2}^{\otimes2} \rangle
            \\
            +
            15 \langle \partial^{3} \vg, \vh_{1}^{\otimes2}\otimes \vh_{4} \rangle
            \\
            +
            60 \langle \partial^{3} \vg, \vh_{1}\otimes \vh_{2}\otimes \vh_{3} \rangle
            \\
            +
            15 \langle \partial^{3} \vg, \vh_{2}^{\otimes3} \rangle
            \\
            +
            6 \langle \partial^{2} \vg, \vh_{1}\otimes \vh_{5} \rangle
            \\
            +
            15 \langle \partial^{2} \vg, \vh_{2}\otimes \vh_{4} \rangle
            \\
            +
            10 \langle \partial^{2} \vg, \vh_{3}^{\otimes2} \rangle
            \\
            +
            \langle \partial \vg, \vh_{6} \rangle
          \end{matrix}
          =
    &\vf_{6}
      =
      \begin{matrix}
        \langle \partial^{6} \vf, \vx_{1}^{\otimes6} \rangle
        \\
        +
        15 \langle \partial^{5} \vf, \vx_{1}^{\otimes4}\otimes \vx_{2} \rangle
        \\
        +
        20 \langle \partial^{4} \vf, \vx_{1}^{\otimes3}\otimes \vx_{3} \rangle
        \\
        +
        45 \langle \partial^{4} \vf, \vx_{1}^{\otimes2}\otimes \vx_{2}^{\otimes2} \rangle
        \\
        +
        15 \langle \partial^{3} \vf, \vx_{1}^{\otimes2}\otimes \vx_{4} \rangle
        \\
        +
        60 \langle \partial^{3} \vf, \vx_{1}\otimes \vx_{2}\otimes \vx_{3} \rangle
        \\
        +
        15 \langle \partial^{3} \vf, \vx_{2}^{\otimes3} \rangle
        \\
        +
        6 \langle \partial^{2} \vf, \vx_{1}\otimes \vx_{5} \rangle
        \\
        +
        15 \langle \partial^{2} \vf, \vx_{2}\otimes \vx_{4} \rangle
        \\
        +
        10 \langle \partial^{2} \vf, \vx_{3}^{\otimes2} \rangle
        \\
        +
        \langle \partial \vf, \vx_{6} \rangle
      \end{matrix}
    \\
    \vx_{7}
    &\to&
          \vh_{7}
          =
          \begin{matrix}
            \langle \partial^{7} \vh, \vx_{1}^{\otimes7} \rangle
            \\
            +
            21 \langle \partial^{6} \vh, \vx_{1}^{\otimes5}\otimes \vx_{2} \rangle
            \\
            +
            35 \langle \partial^{5} \vh, \vx_{1}^{\otimes4}\otimes \vx_{3} \rangle
            \\
            +
            105 \langle \partial^{5} \vh, \vx_{1}^{\otimes3}\otimes \vx_{2}^{\otimes2} \rangle
            \\
            +
            35 \langle \partial^{4} \vh, \vx_{1}^{\otimes3}\otimes \vx_{4} \rangle
            \\
            +
            210 \langle \partial^{4} \vh, \vx_{1}^{\otimes2}\otimes \vx_{2}\otimes \vx_{3} \rangle
            \\
            +
            105 \langle \partial^{4} \vh, \vx_{1}\otimes \vx_{2}^{\otimes3} \rangle
            \\
            +
            21 \langle \partial^{3} \vh, \vx_{1}^{\otimes2}\otimes \vx_{5} \rangle
            \\
            +
            105 \langle \partial^{3} \vh, \vx_{1}\otimes \vx_{2}\otimes \vx_{4} \rangle
            \\
            +
            70 \langle \partial^{3} \vh, \vx_{1}\otimes \vx_{3}^{\otimes2} \rangle
            \\
            +
            105 \langle \partial^{3} \vh, \vx_{2}^{\otimes2}\otimes \vx_{3} \rangle
            \\
            +
            7 \langle \partial^{2} \vh, \vx_{1}\otimes \vx_{6} \rangle
            \\
            +
            21 \langle \partial^{2} \vh, \vx_{2}\otimes \vx_{5} \rangle
            \\
            +
            35 \langle \partial^{2} \vh, \vx_{3}\otimes \vx_{4} \rangle
            \\
            +
            \langle \partial \vh, \vx_{7} \rangle
          \end{matrix}
    &\to&
          \vg_{7}
          =
          \begin{matrix}
            \langle \partial^{7} \vg, \vh_{1}^{\otimes7} \rangle
            \\
            +
            21 \langle \partial^{6} \vg, \vh_{1}^{\otimes5}\otimes \vh_{2} \rangle
            \\
            +
            35 \langle \partial^{5} \vg, \vh_{1}^{\otimes4}\otimes \vh_{3} \rangle
            \\
            +
            105 \langle \partial^{5} \vg, \vh_{1}^{\otimes3}\otimes \vh_{2}^{\otimes2} \rangle
            \\
            +
            35 \langle \partial^{4} \vg, \vh_{1}^{\otimes3}\otimes \vh_{4} \rangle
            \\
            +
            210 \langle \partial^{4} \vg, \vh_{1}^{\otimes2}\otimes \vh_{2}\otimes \vh_{3} \rangle
            \\
            +
            105 \langle \partial^{4} \vg, \vh_{1}\otimes \vh_{2}^{\otimes3} \rangle
            \\
            +
            21 \langle \partial^{3} \vg, \vh_{1}^{\otimes2}\otimes \vh_{5} \rangle
            \\
            +
            105 \langle \partial^{3} \vg, \vh_{1}\otimes \vh_{2}\otimes \vh_{4} \rangle
            \\
            +
            70 \langle \partial^{3} \vg, \vh_{1}\otimes \vh_{3}^{\otimes2} \rangle
            \\
            +
            105 \langle \partial^{3} \vg, \vh_{2}^{\otimes2}\otimes \vh_{3} \rangle
            \\
            +
            7 \langle \partial^{2} \vg, \vh_{1}\otimes \vh_{6} \rangle
            \\
            +
            21 \langle \partial^{2} \vg, \vh_{2}\otimes \vh_{5} \rangle
            \\
            +
            35 \langle \partial^{2} \vg, \vh_{3}\otimes \vh_{4} \rangle
            \\
            +
            \langle \partial \vg, \vh_{7} \rangle
          \end{matrix}
          =
    &\vf_{7}
      =
      \begin{matrix}
        \langle \partial^{7} \vf, \vx_{1}^{\otimes7} \rangle
        \\
        +
        21 \langle \partial^{6} \vf, \vx_{1}^{\otimes5}\otimes \vx_{2} \rangle
        \\
        +
        35 \langle \partial^{5} \vf, \vx_{1}^{\otimes4}\otimes \vx_{3} \rangle
        \\
        +
        105 \langle \partial^{5} \vf, \vx_{1}^{\otimes3}\otimes \vx_{2}^{\otimes2} \rangle
        \\
        +
        35 \langle \partial^{4} \vf, \vx_{1}^{\otimes3}\otimes \vx_{4} \rangle
        \\
        +
        210 \langle \partial^{4} \vf, \vx_{1}^{\otimes2}\otimes \vx_{2}\otimes \vx_{3} \rangle
        \\
        +
        105 \langle \partial^{4} \vf, \vx_{1}\otimes \vx_{2}^{\otimes3} \rangle
        \\
        +
        21 \langle \partial^{3} \vf, \vx_{1}^{\otimes2}\otimes \vx_{5} \rangle
        \\
        +
        105 \langle \partial^{3} \vf, \vx_{1}\otimes \vx_{2}\otimes \vx_{4} \rangle
        \\
        +
        70 \langle \partial^{3} \vf, \vx_{1}\otimes \vx_{3}^{\otimes2} \rangle
        \\
        +
        105 \langle \partial^{3} \vf, \vx_{2}^{\otimes2}\otimes \vx_{3} \rangle
        \\
        +
        7 \langle \partial^{2} \vf, \vx_{1}\otimes \vx_{6} \rangle
        \\
        +
        21 \langle \partial^{2} \vf, \vx_{2}\otimes \vx_{5} \rangle
        \\
        +
        35 \langle \partial^{2} \vf, \vx_{3}\otimes \vx_{4} \rangle
        \\
        +
        \langle \partial \vf, \vx_{7} \rangle
      \end{matrix}
    \\
    \vx_{8}
    &\to&
          \vh_{8}
          =
          \begin{matrix}
            \langle \partial^{8} \vh, \vx_{1}^{\otimes8} \rangle
            \\
            +
            28 \langle \partial^{7} \vh, \vx_{1}^{\otimes6}\otimes \vx_{2} \rangle
            \\
            +
            56 \langle \partial^{6} \vh, \vx_{1}^{\otimes5}\otimes \vx_{3} \rangle
            \\
            +
            210 \langle \partial^{6} \vh, \vx_{1}^{\otimes4}\otimes \vx_{2}^{\otimes2} \rangle
            \\
            +
            70 \langle \partial^{5} \vh, \vx_{1}^{\otimes4}\otimes \vx_{4} \rangle
            \\
            +
            560 \langle \partial^{5} \vh, \vx_{1}^{\otimes3}\otimes \vx_{2}\otimes \vx_{3} \rangle
            \\
            +
            420 \langle \partial^{5} \vh, \vx_{1}^{\otimes2}\otimes \vx_{2}^{\otimes3} \rangle
            \\
            +
            56 \langle \partial^{4} \vh, \vx_{1}^{\otimes3}\otimes \vx_{5} \rangle
            \\
            +
            420 \langle \partial^{4} \vh, \vx_{1}^{\otimes2}\otimes \vx_{2}\otimes \vx_{4} \rangle
            \\
            +
            280 \langle \partial^{4} \vh, \vx_{1}^{\otimes2}\otimes \vx_{3}^{\otimes2} \rangle
            \\
            +
            840 \langle \partial^{4} \vh, \vx_{1}\otimes \vx_{2}^{\otimes2}\otimes \vx_{3} \rangle
            \\
            +
            105 \langle \partial^{4} \vh, \vx_{2}^{\otimes4} \rangle
            \\
            +
            28 \langle \partial^{3} \vh, \vx_{1}^{\otimes2}\otimes \vx_{6} \rangle
            \\
            +
            168 \langle \partial^{3} \vh, \vx_{1}\otimes \vx_{2}\otimes \vx_{5} \rangle
            \\
            +
            280 \langle \partial^{3} \vh, \vx_{1}\otimes \vx_{3}\otimes \vx_{4} \rangle
            \\
            +
            210 \langle \partial^{3} \vh, \vx_{2}^{\otimes2}\otimes \vx_{4} \rangle
            \\
            +
            280 \langle \partial^{3} \vh, \vx_{2}\otimes \vx_{3}^{\otimes2} \rangle
            \\
            +
            8 \langle \partial^{2} \vh, \vx_{1}\otimes \vx_{7} \rangle
            \\
            +
            28 \langle \partial^{2} \vh, \vx_{2}\otimes \vx_{6} \rangle
            \\
            +
            56 \langle \partial^{2} \vh, \vx_{3}\otimes \vx_{5} \rangle
            \\
            +
            35 \langle \partial^{2} \vh, \vx_{4}^{\otimes2} \rangle
            \\
            +
            \langle \partial \vh, \vx_{8} \rangle
          \end{matrix}
    &\to&
          \vg_{8}
          =
          \begin{matrix}
            \langle \partial^{8} \vg, \vh_{1}^{\otimes8} \rangle
            \\
            +
            28 \langle \partial^{7} \vg, \vh_{1}^{\otimes6}\otimes \vh_{2} \rangle
            \\
            +
            56 \langle \partial^{6} \vg, \vh_{1}^{\otimes5}\otimes \vh_{3} \rangle
            \\
            +
            210 \langle \partial^{6} \vg, \vh_{1}^{\otimes4}\otimes \vh_{2}^{\otimes2} \rangle
            \\
            +
            70 \langle \partial^{5} \vg, \vh_{1}^{\otimes4}\otimes \vh_{4} \rangle
            \\
            +
            560 \langle \partial^{5} \vg, \vh_{1}^{\otimes3}\otimes \vh_{2}\otimes \vh_{3} \rangle
            \\
            +
            420 \langle \partial^{5} \vg, \vh_{1}^{\otimes2}\otimes \vh_{2}^{\otimes3} \rangle
            \\
            +
            56 \langle \partial^{4} \vg, \vh_{1}^{\otimes3}\otimes \vh_{5} \rangle
            \\
            +
            420 \langle \partial^{4} \vg, \vh_{1}^{\otimes2}\otimes \vh_{2}\otimes \vh_{4} \rangle
            \\
            +
            280 \langle \partial^{4} \vg, \vh_{1}^{\otimes2}\otimes \vh_{3}^{\otimes2} \rangle
            \\
            +
            840 \langle \partial^{4} \vg, \vh_{1}\otimes \vh_{2}^{\otimes2}\otimes \vh_{3} \rangle
            \\
            +
            105 \langle \partial^{4} \vg, \vh_{2}^{\otimes4} \rangle
            \\
            +
            28 \langle \partial^{3} \vg, \vh_{1}^{\otimes2}\otimes \vh_{6} \rangle
            \\
            +
            168 \langle \partial^{3} \vg, \vh_{1}\otimes \vh_{2}\otimes \vh_{5} \rangle
            \\
            +
            280 \langle \partial^{3} \vg, \vh_{1}\otimes \vh_{3}\otimes \vh_{4} \rangle
            \\
            +
            210 \langle \partial^{3} \vg, \vh_{2}^{\otimes2}\otimes \vh_{4} \rangle
            \\
            +
            280 \langle \partial^{3} \vg, \vh_{2}\otimes \vh_{3}^{\otimes2} \rangle
            \\
            +
            8 \langle \partial^{2} \vg, \vh_{1}\otimes \vh_{7} \rangle
            \\
            +
            28 \langle \partial^{2} \vg, \vh_{2}\otimes \vh_{6} \rangle
            \\
            +
            56 \langle \partial^{2} \vg, \vh_{3}\otimes \vh_{5} \rangle
            \\
            +
            35 \langle \partial^{2} \vg, \vh_{4}^{\otimes2} \rangle
            \\
            +
            \langle \partial \vg, \vh_{8} \rangle
          \end{matrix}
          =
    &\vf_{8}
      =
      \begin{matrix}
        \langle \partial^{8} \vf, \vx_{1}^{\otimes8} \rangle
        \\
        +
        28 \langle \partial^{7} \vf, \vx_{1}^{\otimes6}\otimes \vx_{2} \rangle
        \\
        +
        56 \langle \partial^{6} \vf, \vx_{1}^{\otimes5}\otimes \vx_{3} \rangle
        \\
        +
        210 \langle \partial^{6} \vf, \vx_{1}^{\otimes4}\otimes \vx_{2}^{\otimes2} \rangle
        \\
        +
        70 \langle \partial^{5} \vf, \vx_{1}^{\otimes4}\otimes \vx_{4} \rangle
        \\
        +
        560 \langle \partial^{5} \vf, \vx_{1}^{\otimes3}\otimes \vx_{2}\otimes \vx_{3} \rangle
        \\
        +
        420 \langle \partial^{5} \vf, \vx_{1}^{\otimes2}\otimes \vx_{2}^{\otimes3} \rangle
        \\
        +
        56 \langle \partial^{4} \vf, \vx_{1}^{\otimes3}\otimes \vx_{5} \rangle
        \\
        +
        420 \langle \partial^{4} \vf, \vx_{1}^{\otimes2}\otimes \vx_{2}\otimes \vx_{4} \rangle
        \\
        +
        280 \langle \partial^{4} \vf, \vx_{1}^{\otimes2}\otimes \vx_{3}^{\otimes2} \rangle
        \\
        +
        840 \langle \partial^{4} \vf, \vx_{1}\otimes \vx_{2}^{\otimes2}\otimes \vx_{3} \rangle
        \\
        +
        105 \langle \partial^{4} \vf, \vx_{2}^{\otimes4} \rangle
        \\
        +
        28 \langle \partial^{3} \vf, \vx_{1}^{\otimes2}\otimes \vx_{6} \rangle
        \\
        +
        168 \langle \partial^{3} \vf, \vx_{1}\otimes \vx_{2}\otimes \vx_{5} \rangle
        \\
        +
        280 \langle \partial^{3} \vf, \vx_{1}\otimes \vx_{3}\otimes \vx_{4} \rangle
        \\
        +
        210 \langle \partial^{3} \vf, \vx_{2}^{\otimes2}\otimes \vx_{4} \rangle
        \\
        +
        280 \langle \partial^{3} \vf, \vx_{2}\otimes \vx_{3}^{\otimes2} \rangle
        \\
        +
        8 \langle \partial^{2} \vf, \vx_{1}\otimes \vx_{7} \rangle
        \\
        +
        28 \langle \partial^{2} \vf, \vx_{2}\otimes \vx_{6} \rangle
        \\
        +
        56 \langle \partial^{2} \vf, \vx_{3}\otimes \vx_{5} \rangle
        \\
        +
        35 \langle \partial^{2} \vf, \vx_{4}^{\otimes2} \rangle
        \\
        +
        \langle \partial \vf, \vx_{8} \rangle
      \end{matrix}
  \end{align*}
\end{tiny}
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