There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {e^{x}}^{e^{x + 1}}{x}^{2}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = x^{2}{e^{x}}^{e^{x + 1}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( x^{2}{e^{x}}^{e^{x + 1}}\right)}{dx}\\=&2x{e^{x}}^{e^{x + 1}} + x^{2}({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))\\=&x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1}ln(e^{x}) + x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1} + 2x{e^{x}}^{e^{x + 1}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1}ln(e^{x}) + x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1} + 2x{e^{x}}^{e^{x + 1}}\right)}{dx}\\=&2x{e^{x}}^{e^{x + 1}}e^{x + 1}ln(e^{x}) + x^{2}({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{x + 1}ln(e^{x}) + x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1}(1 + 0)ln(e^{x}) + \frac{x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1}e^{x}}{(e^{x})} + 2x{e^{x}}^{e^{x + 1}}e^{x + 1} + x^{2}({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{x + 1} + x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1}(1 + 0) + 2{e^{x}}^{e^{x + 1}} + 2x({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))\\=&4x{e^{x}}^{e^{x + 1}}e^{x + 1}ln(e^{x}) + x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln^{2}(e^{x}) + 2x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln(e^{x}) + x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1}ln(e^{x}) + 2x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1} + 4x{e^{x}}^{e^{x + 1}}e^{x + 1} + x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}} + 2{e^{x}}^{e^{x + 1}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 4x{e^{x}}^{e^{x + 1}}e^{x + 1}ln(e^{x}) + x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln^{2}(e^{x}) + 2x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln(e^{x}) + x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1}ln(e^{x}) + 2x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1} + 4x{e^{x}}^{e^{x + 1}}e^{x + 1} + x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}} + 2{e^{x}}^{e^{x + 1}}\right)}{dx}\\=&4{e^{x}}^{e^{x + 1}}e^{x + 1}ln(e^{x}) + 4x({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{x + 1}ln(e^{x}) + 4x{e^{x}}^{e^{x + 1}}e^{x + 1}(1 + 0)ln(e^{x}) + \frac{4x{e^{x}}^{e^{x + 1}}e^{x + 1}e^{x}}{(e^{x})} + 2x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln^{2}(e^{x}) + x^{2}({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{{\left(x + 1\right)}*{2}}ln^{2}(e^{x}) + x^{2}{e^{x}}^{e^{x + 1}}*2e^{x + 1}e^{x + 1}(1 + 0)ln^{2}(e^{x}) + \frac{x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}*2ln(e^{x})e^{x}}{(e^{x})} + 2*2x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln(e^{x}) + 2x^{2}({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{{\left(x + 1\right)}*{2}}ln(e^{x}) + 2x^{2}{e^{x}}^{e^{x + 1}}*2e^{x + 1}e^{x + 1}(1 + 0)ln(e^{x}) + \frac{2x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}e^{x}}{(e^{x})} + 2x{e^{x}}^{e^{x + 1}}e^{x + 1}ln(e^{x}) + x^{2}({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{x + 1}ln(e^{x}) + x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1}(1 + 0)ln(e^{x}) + \frac{x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1}e^{x}}{(e^{x})} + 2*2x{e^{x}}^{e^{x + 1}}e^{x + 1} + 2x^{2}({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{x + 1} + 2x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1}(1 + 0) + 4{e^{x}}^{e^{x + 1}}e^{x + 1} + 4x({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{x + 1} + 4x{e^{x}}^{e^{x + 1}}e^{x + 1}(1 + 0) + 2x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}} + x^{2}({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{{\left(x + 1\right)}*{2}} + x^{2}{e^{x}}^{e^{x + 1}}*2e^{x + 1}e^{x + 1}(1 + 0) + 2({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))\\=&6{e^{x}}^{e^{x + 1}}e^{x + 1}ln(e^{x}) + 6x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln^{2}(e^{x}) + x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}}ln^{3}(e^{x}) + 6x{e^{x}}^{e^{x + 1}}e^{x + 1}ln(e^{x}) + 3x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln^{2}(e^{x}) + 3x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}}ln^{2}(e^{x}) + 9x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln(e^{x}) + 12x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln(e^{x}) + x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1}ln(e^{x}) + 3x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}}ln(e^{x}) + 6x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}} + 12x{e^{x}}^{e^{x + 1}}e^{x + 1} + 3x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1} + 6{e^{x}}^{e^{x + 1}}e^{x + 1} + 6x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}} + x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 6{e^{x}}^{e^{x + 1}}e^{x + 1}ln(e^{x}) + 6x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln^{2}(e^{x}) + x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}}ln^{3}(e^{x}) + 6x{e^{x}}^{e^{x + 1}}e^{x + 1}ln(e^{x}) + 3x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln^{2}(e^{x}) + 3x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}}ln^{2}(e^{x}) + 9x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln(e^{x}) + 12x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln(e^{x}) + x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1}ln(e^{x}) + 3x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}}ln(e^{x}) + 6x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}} + 12x{e^{x}}^{e^{x + 1}}e^{x + 1} + 3x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1} + 6{e^{x}}^{e^{x + 1}}e^{x + 1} + 6x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}} + x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}}\right)}{dx}\\=&6({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{x + 1}ln(e^{x}) + 6{e^{x}}^{e^{x + 1}}e^{x + 1}(1 + 0)ln(e^{x}) + \frac{6{e^{x}}^{e^{x + 1}}e^{x + 1}e^{x}}{(e^{x})} + 6{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln^{2}(e^{x}) + 6x({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{{\left(x + 1\right)}*{2}}ln^{2}(e^{x}) + 6x{e^{x}}^{e^{x + 1}}*2e^{x + 1}e^{x + 1}(1 + 0)ln^{2}(e^{x}) + \frac{6x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}*2ln(e^{x})e^{x}}{(e^{x})} + 2x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}}ln^{3}(e^{x}) + x^{2}({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{{\left(x + 1\right)}*{3}}ln^{3}(e^{x}) + x^{2}{e^{x}}^{e^{x + 1}}*3e^{{\left(x + 1\right)}*{2}}e^{x + 1}(1 + 0)ln^{3}(e^{x}) + \frac{x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}}*3ln^{2}(e^{x})e^{x}}{(e^{x})} + 6{e^{x}}^{e^{x + 1}}e^{x + 1}ln(e^{x}) + 6x({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{x + 1}ln(e^{x}) + 6x{e^{x}}^{e^{x + 1}}e^{x + 1}(1 + 0)ln(e^{x}) + \frac{6x{e^{x}}^{e^{x + 1}}e^{x + 1}e^{x}}{(e^{x})} + 3*2x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln^{2}(e^{x}) + 3x^{2}({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{{\left(x + 1\right)}*{2}}ln^{2}(e^{x}) + 3x^{2}{e^{x}}^{e^{x + 1}}*2e^{x + 1}e^{x + 1}(1 + 0)ln^{2}(e^{x}) + \frac{3x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}*2ln(e^{x})e^{x}}{(e^{x})} + 3*2x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}}ln^{2}(e^{x}) + 3x^{2}({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{{\left(x + 1\right)}*{3}}ln^{2}(e^{x}) + 3x^{2}{e^{x}}^{e^{x + 1}}*3e^{{\left(x + 1\right)}*{2}}e^{x + 1}(1 + 0)ln^{2}(e^{x}) + \frac{3x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}}*2ln(e^{x})e^{x}}{(e^{x})} + 9*2x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln(e^{x}) + 9x^{2}({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{{\left(x + 1\right)}*{2}}ln(e^{x}) + 9x^{2}{e^{x}}^{e^{x + 1}}*2e^{x + 1}e^{x + 1}(1 + 0)ln(e^{x}) + \frac{9x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}e^{x}}{(e^{x})} + 12{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln(e^{x}) + 12x({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{{\left(x + 1\right)}*{2}}ln(e^{x}) + 12x{e^{x}}^{e^{x + 1}}*2e^{x + 1}e^{x + 1}(1 + 0)ln(e^{x}) + \frac{12x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}e^{x}}{(e^{x})} + 2x{e^{x}}^{e^{x + 1}}e^{x + 1}ln(e^{x}) + x^{2}({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{x + 1}ln(e^{x}) + x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1}(1 + 0)ln(e^{x}) + \frac{x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1}e^{x}}{(e^{x})} + 3*2x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}}ln(e^{x}) + 3x^{2}({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{{\left(x + 1\right)}*{3}}ln(e^{x}) + 3x^{2}{e^{x}}^{e^{x + 1}}*3e^{{\left(x + 1\right)}*{2}}e^{x + 1}(1 + 0)ln(e^{x}) + \frac{3x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}}e^{x}}{(e^{x})} + 6*2x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}} + 6x^{2}({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{{\left(x + 1\right)}*{2}} + 6x^{2}{e^{x}}^{e^{x + 1}}*2e^{x + 1}e^{x + 1}(1 + 0) + 12{e^{x}}^{e^{x + 1}}e^{x + 1} + 12x({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{x + 1} + 12x{e^{x}}^{e^{x + 1}}e^{x + 1}(1 + 0) + 3*2x{e^{x}}^{e^{x + 1}}e^{x + 1} + 3x^{2}({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{x + 1} + 3x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1}(1 + 0) + 6({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{x + 1} + 6{e^{x}}^{e^{x + 1}}e^{x + 1}(1 + 0) + 6{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}} + 6x({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{{\left(x + 1\right)}*{2}} + 6x{e^{x}}^{e^{x + 1}}*2e^{x + 1}e^{x + 1}(1 + 0) + 2x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}} + x^{2}({e^{x}}^{e^{x + 1}}((e^{x + 1}(1 + 0))ln(e^{x}) + \frac{(e^{x + 1})(e^{x})}{(e^{x})}))e^{{\left(x + 1\right)}*{3}} + x^{2}{e^{x}}^{e^{x + 1}}*3e^{{\left(x + 1\right)}*{2}}e^{x + 1}(1 + 0)\\=&12{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln^{2}(e^{x}) + 24{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln(e^{x}) + 12{e^{x}}^{e^{x + 1}}e^{x + 1}ln(e^{x}) + 8x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}}ln^{3}(e^{x}) + x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{4}}ln^{4}(e^{x}) + 24x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln^{2}(e^{x}) + 72x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln(e^{x}) + 6x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}}ln^{3}(e^{x}) + 24x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}}ln^{2}(e^{x}) + 4x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{4}}ln^{3}(e^{x}) + 8x{e^{x}}^{e^{x + 1}}e^{x + 1}ln(e^{x}) + 7x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln^{2}(e^{x}) + 24x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}}ln^{2}(e^{x}) + 28x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}}ln(e^{x}) + 30x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}}ln(e^{x}) + 6x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{4}}ln^{2}(e^{x}) + x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1}ln(e^{x}) + 24x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}}ln(e^{x}) + 24{e^{x}}^{e^{x + 1}}e^{x + 1} + 4x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{4}}ln(e^{x}) + 24x{e^{x}}^{e^{x + 1}}e^{x + 1} + 4x^{2}{e^{x}}^{e^{x + 1}}e^{x + 1} + 24x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}} + 48x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}} + 12x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}} + 12{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{2}} + 8x{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{3}} + x^{2}{e^{x}}^{e^{x + 1}}e^{{\left(x + 1\right)}*{4}}\\ \end{split}\end{equation} \]

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