There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ xe^{1 - cos(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = xe^{-cos(x) + 1}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( xe^{-cos(x) + 1}\right)}{dx}\\=&e^{-cos(x) + 1} + xe^{-cos(x) + 1}(--sin(x) + 0)\\=&e^{-cos(x) + 1} + xe^{-cos(x) + 1}sin(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( e^{-cos(x) + 1} + xe^{-cos(x) + 1}sin(x)\right)}{dx}\\=&e^{-cos(x) + 1}(--sin(x) + 0) + e^{-cos(x) + 1}sin(x) + xe^{-cos(x) + 1}(--sin(x) + 0)sin(x) + xe^{-cos(x) + 1}cos(x)\\=&2e^{-cos(x) + 1}sin(x) + xe^{-cos(x) + 1}sin^{2}(x) + xe^{-cos(x) + 1}cos(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2e^{-cos(x) + 1}sin(x) + xe^{-cos(x) + 1}sin^{2}(x) + xe^{-cos(x) + 1}cos(x)\right)}{dx}\\=&2e^{-cos(x) + 1}(--sin(x) + 0)sin(x) + 2e^{-cos(x) + 1}cos(x) + e^{-cos(x) + 1}sin^{2}(x) + xe^{-cos(x) + 1}(--sin(x) + 0)sin^{2}(x) + xe^{-cos(x) + 1}*2sin(x)cos(x) + e^{-cos(x) + 1}cos(x) + xe^{-cos(x) + 1}(--sin(x) + 0)cos(x) + xe^{-cos(x) + 1}*-sin(x)\\=&3e^{-cos(x) + 1}sin^{2}(x) + 3e^{-cos(x) + 1}cos(x) + 3xe^{-cos(x) + 1}sin(x)cos(x) + xe^{-cos(x) + 1}sin^{3}(x) - xe^{-cos(x) + 1}sin(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 3e^{-cos(x) + 1}sin^{2}(x) + 3e^{-cos(x) + 1}cos(x) + 3xe^{-cos(x) + 1}sin(x)cos(x) + xe^{-cos(x) + 1}sin^{3}(x) - xe^{-cos(x) + 1}sin(x)\right)}{dx}\\=&3e^{-cos(x) + 1}(--sin(x) + 0)sin^{2}(x) + 3e^{-cos(x) + 1}*2sin(x)cos(x) + 3e^{-cos(x) + 1}(--sin(x) + 0)cos(x) + 3e^{-cos(x) + 1}*-sin(x) + 3e^{-cos(x) + 1}sin(x)cos(x) + 3xe^{-cos(x) + 1}(--sin(x) + 0)sin(x)cos(x) + 3xe^{-cos(x) + 1}cos(x)cos(x) + 3xe^{-cos(x) + 1}sin(x)*-sin(x) + e^{-cos(x) + 1}sin^{3}(x) + xe^{-cos(x) + 1}(--sin(x) + 0)sin^{3}(x) + xe^{-cos(x) + 1}*3sin^{2}(x)cos(x) - e^{-cos(x) + 1}sin(x) - xe^{-cos(x) + 1}(--sin(x) + 0)sin(x) - xe^{-cos(x) + 1}cos(x)\\=&12e^{-cos(x) + 1}sin(x)cos(x) + 4e^{-cos(x) + 1}sin^{3}(x) - 4e^{-cos(x) + 1}sin(x) + 6xe^{-cos(x) + 1}sin^{2}(x)cos(x) + 3xe^{-cos(x) + 1}cos^{2}(x) - 4xe^{-cos(x) + 1}sin^{2}(x) + xe^{-cos(x) + 1}sin^{4}(x) - xe^{-cos(x) + 1}cos(x)\\ \end{split}\end{equation} \]

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