There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ e^{{(x - 1)}^{2}{(x + 1)}^{3}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}\right)}{dx}\\=&e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0)\\=&5x^{4}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 4x^{3}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 6x^{2}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 4xe^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 5x^{4}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 4x^{3}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 6x^{2}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 4xe^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}\right)}{dx}\\=&5*4x^{3}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 5x^{4}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) + 4*3x^{2}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 4x^{3}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) - 6*2xe^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 6x^{2}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) - 4e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 4xe^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) + e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0)\\=&76x^{3}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 25x^{8}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 40x^{7}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 44x^{6}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 88x^{5}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 14x^{4}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 16x^{2}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 20xe^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 3e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 76x^{3}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 25x^{8}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 40x^{7}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 44x^{6}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 88x^{5}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 14x^{4}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 16x^{2}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 20xe^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 3e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}\right)}{dx}\\=&76*3x^{2}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 76x^{3}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) + 25*8x^{7}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 25x^{8}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) + 40*7x^{6}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 40x^{7}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) - 44*6x^{5}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 44x^{6}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) - 88*5x^{4}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 88x^{5}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) + 14*4x^{3}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 14x^{4}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) + 16*2xe^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 16x^{2}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) - 20e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 20xe^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) - 3e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0)\\=&342x^{2}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 1380x^{7}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 888x^{6}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 900x^{5}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 921x^{4}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 176x^{3}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 125x^{12}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 300x^{11}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 210x^{10}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 956x^{9}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 153x^{8}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 24xe^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 23e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 342x^{2}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 1380x^{7}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 888x^{6}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 900x^{5}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 921x^{4}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 176x^{3}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 125x^{12}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 300x^{11}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 210x^{10}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 956x^{9}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 153x^{8}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 24xe^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 23e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}\right)}{dx}\\=&342*2xe^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 342x^{2}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) + 1380*7x^{6}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 1380x^{7}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) + 888*6x^{5}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 888x^{6}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) - 900*5x^{4}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 900x^{5}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) - 921*4x^{3}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 921x^{4}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) + 176*3x^{2}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 176x^{3}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) + 125*12x^{11}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 125x^{12}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) + 300*11x^{10}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 300x^{11}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) - 210*10x^{9}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 210x^{10}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) - 956*9x^{8}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 956x^{9}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) - 153*8x^{7}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 153x^{8}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) + 24e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 24xe^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0) - 23e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}(5x^{4} + 4x^{3} - 2*3x^{2} - 2*2x + 1 + 0)\\=&800xe^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 22088x^{6}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 8544x^{5}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 8196x^{4}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 5112x^{3}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 912x^{2}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 14664x^{11}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 17792x^{10}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 11672x^{9}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 27810x^{8}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 800x^{7}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 625x^{16}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + 2000x^{15}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 600x^{14}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 7920x^{13}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} - 4404x^{12}e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1} + e^{x^{5} + x^{4} - 2x^{3} - 2x^{2} + x + 1}\\ \end{split}\end{equation} \]

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