There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ e^{{x}^{x}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{{x}^{x}}\right)}{dx}\\=&e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))\\=&{x}^{x}e^{{x}^{x}}ln(x) + {x}^{x}e^{{x}^{x}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( {x}^{x}e^{{x}^{x}}ln(x) + {x}^{x}e^{{x}^{x}}\right)}{dx}\\=&({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))e^{{x}^{x}}ln(x) + {x}^{x}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(x) + \frac{{x}^{x}e^{{x}^{x}}}{(x)} + ({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))e^{{x}^{x}} + {x}^{x}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))\\=&{x}^{x}e^{{x}^{x}}ln^{2}(x) + {x}^{(2x)}e^{{x}^{x}}ln^{2}(x) + 2{x}^{x}e^{{x}^{x}}ln(x) + 2{x}^{(2x)}e^{{x}^{x}}ln(x) + \frac{{x}^{x}e^{{x}^{x}}}{x} + {x}^{x}e^{{x}^{x}} + {x}^{(2x)}e^{{x}^{x}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( {x}^{x}e^{{x}^{x}}ln^{2}(x) + {x}^{(2x)}e^{{x}^{x}}ln^{2}(x) + 2{x}^{x}e^{{x}^{x}}ln(x) + 2{x}^{(2x)}e^{{x}^{x}}ln(x) + \frac{{x}^{x}e^{{x}^{x}}}{x} + {x}^{x}e^{{x}^{x}} + {x}^{(2x)}e^{{x}^{x}}\right)}{dx}\\=&({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))e^{{x}^{x}}ln^{2}(x) + {x}^{x}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln^{2}(x) + \frac{{x}^{x}e^{{x}^{x}}*2ln(x)}{(x)} + ({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))e^{{x}^{x}}ln^{2}(x) + {x}^{(2x)}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln^{2}(x) + \frac{{x}^{(2x)}e^{{x}^{x}}*2ln(x)}{(x)} + 2({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))e^{{x}^{x}}ln(x) + 2{x}^{x}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(x) + \frac{2{x}^{x}e^{{x}^{x}}}{(x)} + 2({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))e^{{x}^{x}}ln(x) + 2{x}^{(2x)}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(x) + \frac{2{x}^{(2x)}e^{{x}^{x}}}{(x)} + \frac{-{x}^{x}e^{{x}^{x}}}{x^{2}} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))e^{{x}^{x}}}{x} + \frac{{x}^{x}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{x} + ({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))e^{{x}^{x}} + {x}^{x}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + ({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))e^{{x}^{x}} + {x}^{(2x)}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))\\=&{x}^{x}e^{{x}^{x}}ln^{3}(x) + 3{x}^{(2x)}e^{{x}^{x}}ln^{3}(x) + {x}^{(3x)}e^{{x}^{x}}ln^{3}(x) + \frac{3{x}^{x}e^{{x}^{x}}ln(x)}{x} + 3{x}^{x}e^{{x}^{x}}ln^{2}(x) + 9{x}^{(2x)}e^{{x}^{x}}ln^{2}(x) + 3{x}^{(3x)}e^{{x}^{x}}ln^{2}(x) + \frac{3{x}^{(2x)}e^{{x}^{x}}ln(x)}{x} + 3{x}^{x}e^{{x}^{x}}ln(x) + 9{x}^{(2x)}e^{{x}^{x}}ln(x) + 3{x}^{(3x)}e^{{x}^{x}}ln(x) + \frac{3{x}^{(2x)}e^{{x}^{x}}}{x} - \frac{{x}^{x}e^{{x}^{x}}}{x^{2}} + \frac{3{x}^{x}e^{{x}^{x}}}{x} + {x}^{x}e^{{x}^{x}} + 3{x}^{(2x)}e^{{x}^{x}} + {x}^{(3x)}e^{{x}^{x}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( {x}^{x}e^{{x}^{x}}ln^{3}(x) + 3{x}^{(2x)}e^{{x}^{x}}ln^{3}(x) + {x}^{(3x)}e^{{x}^{x}}ln^{3}(x) + \frac{3{x}^{x}e^{{x}^{x}}ln(x)}{x} + 3{x}^{x}e^{{x}^{x}}ln^{2}(x) + 9{x}^{(2x)}e^{{x}^{x}}ln^{2}(x) + 3{x}^{(3x)}e^{{x}^{x}}ln^{2}(x) + \frac{3{x}^{(2x)}e^{{x}^{x}}ln(x)}{x} + 3{x}^{x}e^{{x}^{x}}ln(x) + 9{x}^{(2x)}e^{{x}^{x}}ln(x) + 3{x}^{(3x)}e^{{x}^{x}}ln(x) + \frac{3{x}^{(2x)}e^{{x}^{x}}}{x} - \frac{{x}^{x}e^{{x}^{x}}}{x^{2}} + \frac{3{x}^{x}e^{{x}^{x}}}{x} + {x}^{x}e^{{x}^{x}} + 3{x}^{(2x)}e^{{x}^{x}} + {x}^{(3x)}e^{{x}^{x}}\right)}{dx}\\=&({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))e^{{x}^{x}}ln^{3}(x) + {x}^{x}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln^{3}(x) + \frac{{x}^{x}e^{{x}^{x}}*3ln^{2}(x)}{(x)} + 3({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))e^{{x}^{x}}ln^{3}(x) + 3{x}^{(2x)}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln^{3}(x) + \frac{3{x}^{(2x)}e^{{x}^{x}}*3ln^{2}(x)}{(x)} + ({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)}))e^{{x}^{x}}ln^{3}(x) + {x}^{(3x)}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln^{3}(x) + \frac{{x}^{(3x)}e^{{x}^{x}}*3ln^{2}(x)}{(x)} + \frac{3*-{x}^{x}e^{{x}^{x}}ln(x)}{x^{2}} + \frac{3({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))e^{{x}^{x}}ln(x)}{x} + \frac{3{x}^{x}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(x)}{x} + \frac{3{x}^{x}e^{{x}^{x}}}{x(x)} + 3({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))e^{{x}^{x}}ln^{2}(x) + 3{x}^{x}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln^{2}(x) + \frac{3{x}^{x}e^{{x}^{x}}*2ln(x)}{(x)} + 9({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))e^{{x}^{x}}ln^{2}(x) + 9{x}^{(2x)}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln^{2}(x) + \frac{9{x}^{(2x)}e^{{x}^{x}}*2ln(x)}{(x)} + 3({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)}))e^{{x}^{x}}ln^{2}(x) + 3{x}^{(3x)}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln^{2}(x) + \frac{3{x}^{(3x)}e^{{x}^{x}}*2ln(x)}{(x)} + \frac{3*-{x}^{(2x)}e^{{x}^{x}}ln(x)}{x^{2}} + \frac{3({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))e^{{x}^{x}}ln(x)}{x} + \frac{3{x}^{(2x)}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(x)}{x} + \frac{3{x}^{(2x)}e^{{x}^{x}}}{x(x)} + 3({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))e^{{x}^{x}}ln(x) + 3{x}^{x}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(x) + \frac{3{x}^{x}e^{{x}^{x}}}{(x)} + 9({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))e^{{x}^{x}}ln(x) + 9{x}^{(2x)}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(x) + \frac{9{x}^{(2x)}e^{{x}^{x}}}{(x)} + 3({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)}))e^{{x}^{x}}ln(x) + 3{x}^{(3x)}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(x) + \frac{3{x}^{(3x)}e^{{x}^{x}}}{(x)} + \frac{3*-{x}^{(2x)}e^{{x}^{x}}}{x^{2}} + \frac{3({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))e^{{x}^{x}}}{x} + \frac{3{x}^{(2x)}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{x} - \frac{-2{x}^{x}e^{{x}^{x}}}{x^{3}} - \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))e^{{x}^{x}}}{x^{2}} - \frac{{x}^{x}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{x^{2}} + \frac{3*-{x}^{x}e^{{x}^{x}}}{x^{2}} + \frac{3({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))e^{{x}^{x}}}{x} + \frac{3{x}^{x}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{x} + ({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))e^{{x}^{x}} + {x}^{x}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 3({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))e^{{x}^{x}} + 3{x}^{(2x)}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + ({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)}))e^{{x}^{x}} + {x}^{(3x)}e^{{x}^{x}}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))\\=&{x}^{x}e^{{x}^{x}}ln^{4}(x) + 7{x}^{(2x)}e^{{x}^{x}}ln^{4}(x) + 6{x}^{(3x)}e^{{x}^{x}}ln^{4}(x) + \frac{6{x}^{x}e^{{x}^{x}}ln^{2}(x)}{x} + {x}^{(4x)}e^{{x}^{x}}ln^{4}(x) + 4{x}^{x}e^{{x}^{x}}ln^{3}(x) + 28{x}^{(2x)}e^{{x}^{x}}ln^{3}(x) + \frac{18{x}^{(2x)}e^{{x}^{x}}ln^{2}(x)}{x} + 24{x}^{(3x)}e^{{x}^{x}}ln^{3}(x) + 4{x}^{(4x)}e^{{x}^{x}}ln^{3}(x) + \frac{6{x}^{(3x)}e^{{x}^{x}}ln^{2}(x)}{x} - \frac{4{x}^{x}e^{{x}^{x}}ln(x)}{x^{2}} + 6{x}^{x}e^{{x}^{x}}ln^{2}(x) + 42{x}^{(2x)}e^{{x}^{x}}ln^{2}(x) + 36{x}^{(3x)}e^{{x}^{x}}ln^{2}(x) + \frac{12{x}^{x}e^{{x}^{x}}ln(x)}{x} + 6{x}^{(4x)}e^{{x}^{x}}ln^{2}(x) + \frac{36{x}^{(2x)}e^{{x}^{x}}ln(x)}{x} + \frac{12{x}^{(3x)}e^{{x}^{x}}ln(x)}{x} - \frac{4{x}^{(2x)}e^{{x}^{x}}ln(x)}{x^{2}} + 4{x}^{x}e^{{x}^{x}}ln(x) + 28{x}^{(2x)}e^{{x}^{x}}ln(x) + 24{x}^{(3x)}e^{{x}^{x}}ln(x) + \frac{18{x}^{(2x)}e^{{x}^{x}}}{x} + 4{x}^{(4x)}e^{{x}^{x}}ln(x) + \frac{6{x}^{(3x)}e^{{x}^{x}}}{x} + \frac{6{x}^{x}e^{{x}^{x}}}{x} + \frac{2{x}^{x}e^{{x}^{x}}}{x^{3}} - \frac{{x}^{(2x)}e^{{x}^{x}}}{x^{2}} - \frac{{x}^{x}e^{{x}^{x}}}{x^{2}} + 7{x}^{(2x)}e^{{x}^{x}} + 6{x}^{(3x)}e^{{x}^{x}} + {x}^{x}e^{{x}^{x}} + {x}^{(4x)}e^{{x}^{x}}\\ \end{split}\end{equation} \]

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