There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{({x}^{x} - 2)}{(2{(x - 1)}^{x})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{1}{2} - (x - 1)^{(-x)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{1}{2} - (x - 1)^{(-x)}\right)}{dx}\\=&0 - ((x - 1)^{(-x)}((-1)ln(x - 1) + \frac{(-x)(1 + 0)}{(x - 1)}))\\=&(x - 1)^{(-x)}ln(x - 1) + \frac{x(x - 1)^{(-x)}}{(x - 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( (x - 1)^{(-x)}ln(x - 1) + \frac{x(x - 1)^{(-x)}}{(x - 1)}\right)}{dx}\\=&((x - 1)^{(-x)}((-1)ln(x - 1) + \frac{(-x)(1 + 0)}{(x - 1)}))ln(x - 1) + \frac{(x - 1)^{(-x)}(1 + 0)}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}})x(x - 1)^{(-x)} + \frac{(x - 1)^{(-x)}}{(x - 1)} + \frac{x((x - 1)^{(-x)}((-1)ln(x - 1) + \frac{(-x)(1 + 0)}{(x - 1)}))}{(x - 1)}\\=&-(x - 1)^{(-x)}ln^{2}(x - 1) - \frac{2x(x - 1)^{(-x)}ln(x - 1)}{(x - 1)} + \frac{2(x - 1)^{(-x)}}{(x - 1)} - \frac{x(x - 1)^{(-x)}}{(x - 1)^{2}} - \frac{x^{2}(x - 1)^{(-x)}}{(x - 1)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -(x - 1)^{(-x)}ln^{2}(x - 1) - \frac{2x(x - 1)^{(-x)}ln(x - 1)}{(x - 1)} + \frac{2(x - 1)^{(-x)}}{(x - 1)} - \frac{x(x - 1)^{(-x)}}{(x - 1)^{2}} - \frac{x^{2}(x - 1)^{(-x)}}{(x - 1)^{2}}\right)}{dx}\\=&-((x - 1)^{(-x)}((-1)ln(x - 1) + \frac{(-x)(1 + 0)}{(x - 1)}))ln^{2}(x - 1) - \frac{(x - 1)^{(-x)}*2ln(x - 1)(1 + 0)}{(x - 1)} - 2(\frac{-(1 + 0)}{(x - 1)^{2}})x(x - 1)^{(-x)}ln(x - 1) - \frac{2(x - 1)^{(-x)}ln(x - 1)}{(x - 1)} - \frac{2x((x - 1)^{(-x)}((-1)ln(x - 1) + \frac{(-x)(1 + 0)}{(x - 1)}))ln(x - 1)}{(x - 1)} - \frac{2x(x - 1)^{(-x)}(1 + 0)}{(x - 1)(x - 1)} + 2(\frac{-(1 + 0)}{(x - 1)^{2}})(x - 1)^{(-x)} + \frac{2((x - 1)^{(-x)}((-1)ln(x - 1) + \frac{(-x)(1 + 0)}{(x - 1)}))}{(x - 1)} - (\frac{-2(1 + 0)}{(x - 1)^{3}})x(x - 1)^{(-x)} - \frac{(x - 1)^{(-x)}}{(x - 1)^{2}} - \frac{x((x - 1)^{(-x)}((-1)ln(x - 1) + \frac{(-x)(1 + 0)}{(x - 1)}))}{(x - 1)^{2}} - (\frac{-2(1 + 0)}{(x - 1)^{3}})x^{2}(x - 1)^{(-x)} - \frac{2x(x - 1)^{(-x)}}{(x - 1)^{2}} - \frac{x^{2}((x - 1)^{(-x)}((-1)ln(x - 1) + \frac{(-x)(1 + 0)}{(x - 1)}))}{(x - 1)^{2}}\\=&(x - 1)^{(-x)}ln^{3}(x - 1) + \frac{3x(x - 1)^{(-x)}ln^{2}(x - 1)}{(x - 1)} - \frac{6(x - 1)^{(-x)}ln(x - 1)}{(x - 1)} + \frac{3x(x - 1)^{(-x)}ln(x - 1)}{(x - 1)^{2}} + \frac{3x^{2}(x - 1)^{(-x)}ln(x - 1)}{(x - 1)^{2}} - \frac{6x(x - 1)^{(-x)}}{(x - 1)^{2}} - \frac{3(x - 1)^{(-x)}}{(x - 1)^{2}} + \frac{2x(x - 1)^{(-x)}}{(x - 1)^{3}} + \frac{3x^{2}(x - 1)^{(-x)}}{(x - 1)^{3}} + \frac{x^{3}(x - 1)^{(-x)}}{(x - 1)^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( (x - 1)^{(-x)}ln^{3}(x - 1) + \frac{3x(x - 1)^{(-x)}ln^{2}(x - 1)}{(x - 1)} - \frac{6(x - 1)^{(-x)}ln(x - 1)}{(x - 1)} + \frac{3x(x - 1)^{(-x)}ln(x - 1)}{(x - 1)^{2}} + \frac{3x^{2}(x - 1)^{(-x)}ln(x - 1)}{(x - 1)^{2}} - \frac{6x(x - 1)^{(-x)}}{(x - 1)^{2}} - \frac{3(x - 1)^{(-x)}}{(x - 1)^{2}} + \frac{2x(x - 1)^{(-x)}}{(x - 1)^{3}} + \frac{3x^{2}(x - 1)^{(-x)}}{(x - 1)^{3}} + \frac{x^{3}(x - 1)^{(-x)}}{(x - 1)^{3}}\right)}{dx}\\=&((x - 1)^{(-x)}((-1)ln(x - 1) + \frac{(-x)(1 + 0)}{(x - 1)}))ln^{3}(x - 1) + \frac{(x - 1)^{(-x)}*3ln^{2}(x - 1)(1 + 0)}{(x - 1)} + 3(\frac{-(1 + 0)}{(x - 1)^{2}})x(x - 1)^{(-x)}ln^{2}(x - 1) + \frac{3(x - 1)^{(-x)}ln^{2}(x - 1)}{(x - 1)} + \frac{3x((x - 1)^{(-x)}((-1)ln(x - 1) + \frac{(-x)(1 + 0)}{(x - 1)}))ln^{2}(x - 1)}{(x - 1)} + \frac{3x(x - 1)^{(-x)}*2ln(x - 1)(1 + 0)}{(x - 1)(x - 1)} - 6(\frac{-(1 + 0)}{(x - 1)^{2}})(x - 1)^{(-x)}ln(x - 1) - \frac{6((x - 1)^{(-x)}((-1)ln(x - 1) + \frac{(-x)(1 + 0)}{(x - 1)}))ln(x - 1)}{(x - 1)} - \frac{6(x - 1)^{(-x)}(1 + 0)}{(x - 1)(x - 1)} + 3(\frac{-2(1 + 0)}{(x - 1)^{3}})x(x - 1)^{(-x)}ln(x - 1) + \frac{3(x - 1)^{(-x)}ln(x - 1)}{(x - 1)^{2}} + \frac{3x((x - 1)^{(-x)}((-1)ln(x - 1) + \frac{(-x)(1 + 0)}{(x - 1)}))ln(x - 1)}{(x - 1)^{2}} + \frac{3x(x - 1)^{(-x)}(1 + 0)}{(x - 1)^{2}(x - 1)} + 3(\frac{-2(1 + 0)}{(x - 1)^{3}})x^{2}(x - 1)^{(-x)}ln(x - 1) + \frac{3*2x(x - 1)^{(-x)}ln(x - 1)}{(x - 1)^{2}} + \frac{3x^{2}((x - 1)^{(-x)}((-1)ln(x - 1) + \frac{(-x)(1 + 0)}{(x - 1)}))ln(x - 1)}{(x - 1)^{2}} + \frac{3x^{2}(x - 1)^{(-x)}(1 + 0)}{(x - 1)^{2}(x - 1)} - 6(\frac{-2(1 + 0)}{(x - 1)^{3}})x(x - 1)^{(-x)} - \frac{6(x - 1)^{(-x)}}{(x - 1)^{2}} - \frac{6x((x - 1)^{(-x)}((-1)ln(x - 1) + \frac{(-x)(1 + 0)}{(x - 1)}))}{(x - 1)^{2}} - 3(\frac{-2(1 + 0)}{(x - 1)^{3}})(x - 1)^{(-x)} - \frac{3((x - 1)^{(-x)}((-1)ln(x - 1) + \frac{(-x)(1 + 0)}{(x - 1)}))}{(x - 1)^{2}} + 2(\frac{-3(1 + 0)}{(x - 1)^{4}})x(x - 1)^{(-x)} + \frac{2(x - 1)^{(-x)}}{(x - 1)^{3}} + \frac{2x((x - 1)^{(-x)}((-1)ln(x - 1) + \frac{(-x)(1 + 0)}{(x - 1)}))}{(x - 1)^{3}} + 3(\frac{-3(1 + 0)}{(x - 1)^{4}})x^{2}(x - 1)^{(-x)} + \frac{3*2x(x - 1)^{(-x)}}{(x - 1)^{3}} + \frac{3x^{2}((x - 1)^{(-x)}((-1)ln(x - 1) + \frac{(-x)(1 + 0)}{(x - 1)}))}{(x - 1)^{3}} + (\frac{-3(1 + 0)}{(x - 1)^{4}})x^{3}(x - 1)^{(-x)} + \frac{3x^{2}(x - 1)^{(-x)}}{(x - 1)^{3}} + \frac{x^{3}((x - 1)^{(-x)}((-1)ln(x - 1) + \frac{(-x)(1 + 0)}{(x - 1)}))}{(x - 1)^{3}}\\=&-(x - 1)^{(-x)}ln^{4}(x - 1) - \frac{4x(x - 1)^{(-x)}ln^{3}(x - 1)}{(x - 1)} + \frac{12(x - 1)^{(-x)}ln^{2}(x - 1)}{(x - 1)} - \frac{6x(x - 1)^{(-x)}ln^{2}(x - 1)}{(x - 1)^{2}} - \frac{6x^{2}(x - 1)^{(-x)}ln^{2}(x - 1)}{(x - 1)^{2}} + \frac{24x(x - 1)^{(-x)}ln(x - 1)}{(x - 1)^{2}} + \frac{12(x - 1)^{(-x)}ln(x - 1)}{(x - 1)^{2}} - \frac{12(x - 1)^{(-x)}}{(x - 1)^{2}} - \frac{8x(x - 1)^{(-x)}ln(x - 1)}{(x - 1)^{3}} - \frac{12x^{2}(x - 1)^{(-x)}ln(x - 1)}{(x - 1)^{3}} - \frac{4x^{3}(x - 1)^{(-x)}ln(x - 1)}{(x - 1)^{3}} + \frac{24x(x - 1)^{(-x)}}{(x - 1)^{3}} + \frac{12x^{2}(x - 1)^{(-x)}}{(x - 1)^{3}} + \frac{8(x - 1)^{(-x)}}{(x - 1)^{3}} - \frac{6x(x - 1)^{(-x)}}{(x - 1)^{4}} - \frac{11x^{2}(x - 1)^{(-x)}}{(x - 1)^{4}} - \frac{6x^{3}(x - 1)^{(-x)}}{(x - 1)^{4}} - \frac{x^{4}(x - 1)^{(-x)}}{(x - 1)^{4}}\\ \end{split}\end{equation} \]

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