本次共计算 1 个题目：每一题对 x 求 4 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数e^{axxx + bxx + cx + d} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = e^{ax^{3} + bx^{2} + cx + d}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( e^{ax^{3} + bx^{2} + cx + d}\right)}{dx}\\=&e^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0)\\=&3ax^{2}e^{ax^{3} + bx^{2} + cx + d} + 2bxe^{ax^{3} + bx^{2} + cx + d} + ce^{ax^{3} + bx^{2} + cx + d}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( 3ax^{2}e^{ax^{3} + bx^{2} + cx + d} + 2bxe^{ax^{3} + bx^{2} + cx + d} + ce^{ax^{3} + bx^{2} + cx + d}\right)}{dx}\\=&3a*2xe^{ax^{3} + bx^{2} + cx + d} + 3ax^{2}e^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 2be^{ax^{3} + bx^{2} + cx + d} + 2bxe^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + ce^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0)\\=&6axe^{ax^{3} + bx^{2} + cx + d} + 9a^{2}x^{4}e^{ax^{3} + bx^{2} + cx + d} + 12abx^{3}e^{ax^{3} + bx^{2} + cx + d} + 6acx^{2}e^{ax^{3} + bx^{2} + cx + d} + 2be^{ax^{3} + bx^{2} + cx + d} + 4b^{2}x^{2}e^{ax^{3} + bx^{2} + cx + d} + 4bcxe^{ax^{3} + bx^{2} + cx + d} + c^{2}e^{ax^{3} + bx^{2} + cx + d}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( 6axe^{ax^{3} + bx^{2} + cx + d} + 9a^{2}x^{4}e^{ax^{3} + bx^{2} + cx + d} + 12abx^{3}e^{ax^{3} + bx^{2} + cx + d} + 6acx^{2}e^{ax^{3} + bx^{2} + cx + d} + 2be^{ax^{3} + bx^{2} + cx + d} + 4b^{2}x^{2}e^{ax^{3} + bx^{2} + cx + d} + 4bcxe^{ax^{3} + bx^{2} + cx + d} + c^{2}e^{ax^{3} + bx^{2} + cx + d}\right)}{dx}\\=&6ae^{ax^{3} + bx^{2} + cx + d} + 6axe^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 9a^{2}*4x^{3}e^{ax^{3} + bx^{2} + cx + d} + 9a^{2}x^{4}e^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 12ab*3x^{2}e^{ax^{3} + bx^{2} + cx + d} + 12abx^{3}e^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 6ac*2xe^{ax^{3} + bx^{2} + cx + d} + 6acx^{2}e^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 2be^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 4b^{2}*2xe^{ax^{3} + bx^{2} + cx + d} + 4b^{2}x^{2}e^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 4bce^{ax^{3} + bx^{2} + cx + d} + 4bcxe^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + c^{2}e^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0)\\=&6ae^{ax^{3} + bx^{2} + cx + d} + 54a^{2}x^{3}e^{ax^{3} + bx^{2} + cx + d} + 54abx^{2}e^{ax^{3} + bx^{2} + cx + d} + 18acxe^{ax^{3} + bx^{2} + cx + d} + 27a^{3}x^{6}e^{ax^{3} + bx^{2} + cx + d} + 54a^{2}bx^{5}e^{ax^{3} + bx^{2} + cx + d} + 27a^{2}cx^{4}e^{ax^{3} + bx^{2} + cx + d} + 36ab^{2}x^{4}e^{ax^{3} + bx^{2} + cx + d} + 36abcx^{3}e^{ax^{3} + bx^{2} + cx + d} + 9ac^{2}x^{2}e^{ax^{3} + bx^{2} + cx + d} + 12b^{2}xe^{ax^{3} + bx^{2} + cx + d} + 6bce^{ax^{3} + bx^{2} + cx + d} + 8b^{3}x^{3}e^{ax^{3} + bx^{2} + cx + d} + 12b^{2}cx^{2}e^{ax^{3} + bx^{2} + cx + d} + 6bc^{2}xe^{ax^{3} + bx^{2} + cx + d} + c^{3}e^{ax^{3} + bx^{2} + cx + d}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( 6ae^{ax^{3} + bx^{2} + cx + d} + 54a^{2}x^{3}e^{ax^{3} + bx^{2} + cx + d} + 54abx^{2}e^{ax^{3} + bx^{2} + cx + d} + 18acxe^{ax^{3} + bx^{2} + cx + d} + 27a^{3}x^{6}e^{ax^{3} + bx^{2} + cx + d} + 54a^{2}bx^{5}e^{ax^{3} + bx^{2} + cx + d} + 27a^{2}cx^{4}e^{ax^{3} + bx^{2} + cx + d} + 36ab^{2}x^{4}e^{ax^{3} + bx^{2} + cx + d} + 36abcx^{3}e^{ax^{3} + bx^{2} + cx + d} + 9ac^{2}x^{2}e^{ax^{3} + bx^{2} + cx + d} + 12b^{2}xe^{ax^{3} + bx^{2} + cx + d} + 6bce^{ax^{3} + bx^{2} + cx + d} + 8b^{3}x^{3}e^{ax^{3} + bx^{2} + cx + d} + 12b^{2}cx^{2}e^{ax^{3} + bx^{2} + cx + d} + 6bc^{2}xe^{ax^{3} + bx^{2} + cx + d} + c^{3}e^{ax^{3} + bx^{2} + cx + d}\right)}{dx}\\=&6ae^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 54a^{2}*3x^{2}e^{ax^{3} + bx^{2} + cx + d} + 54a^{2}x^{3}e^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 54ab*2xe^{ax^{3} + bx^{2} + cx + d} + 54abx^{2}e^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 18ace^{ax^{3} + bx^{2} + cx + d} + 18acxe^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 27a^{3}*6x^{5}e^{ax^{3} + bx^{2} + cx + d} + 27a^{3}x^{6}e^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 54a^{2}b*5x^{4}e^{ax^{3} + bx^{2} + cx + d} + 54a^{2}bx^{5}e^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 27a^{2}c*4x^{3}e^{ax^{3} + bx^{2} + cx + d} + 27a^{2}cx^{4}e^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 36ab^{2}*4x^{3}e^{ax^{3} + bx^{2} + cx + d} + 36ab^{2}x^{4}e^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 36abc*3x^{2}e^{ax^{3} + bx^{2} + cx + d} + 36abcx^{3}e^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 9ac^{2}*2xe^{ax^{3} + bx^{2} + cx + d} + 9ac^{2}x^{2}e^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 12b^{2}e^{ax^{3} + bx^{2} + cx + d} + 12b^{2}xe^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 6bce^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 8b^{3}*3x^{2}e^{ax^{3} + bx^{2} + cx + d} + 8b^{3}x^{3}e^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 12b^{2}c*2xe^{ax^{3} + bx^{2} + cx + d} + 12b^{2}cx^{2}e^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + 6bc^{2}e^{ax^{3} + bx^{2} + cx + d} + 6bc^{2}xe^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0) + c^{3}e^{ax^{3} + bx^{2} + cx + d}(a*3x^{2} + b*2x + c + 0)\\=&180a^{2}x^{2}e^{ax^{3} + bx^{2} + cx + d} + 120abxe^{ax^{3} + bx^{2} + cx + d} + 24ace^{ax^{3} + bx^{2} + cx + d} + 324a^{3}x^{5}e^{ax^{3} + bx^{2} + cx + d} + 540a^{2}bx^{4}e^{ax^{3} + bx^{2} + cx + d} + 216a^{2}cx^{3}e^{ax^{3} + bx^{2} + cx + d} + 288ab^{2}x^{3}e^{ax^{3} + bx^{2} + cx + d} + 216abcx^{2}e^{ax^{3} + bx^{2} + cx + d} + 36ac^{2}xe^{ax^{3} + bx^{2} + cx + d} + 81a^{4}x^{8}e^{ax^{3} + bx^{2} + cx + d} + 216a^{3}bx^{7}e^{ax^{3} + bx^{2} + cx + d} + 108a^{3}cx^{6}e^{ax^{3} + bx^{2} + cx + d} + 216a^{2}b^{2}x^{6}e^{ax^{3} + bx^{2} + cx + d} + 216a^{2}bcx^{5}e^{ax^{3} + bx^{2} + cx + d} + 54a^{2}c^{2}x^{4}e^{ax^{3} + bx^{2} + cx + d} + 96ab^{3}x^{5}e^{ax^{3} + bx^{2} + cx + d} + 144ab^{2}cx^{4}e^{ax^{3} + bx^{2} + cx + d} + 72abc^{2}x^{3}e^{ax^{3} + bx^{2} + cx + d} + 12ac^{3}x^{2}e^{ax^{3} + bx^{2} + cx + d} + 12b^{2}e^{ax^{3} + bx^{2} + cx + d} + 48b^{3}x^{2}e^{ax^{3} + bx^{2} + cx + d} + 48b^{2}cxe^{ax^{3} + bx^{2} + cx + d} + 12bc^{2}e^{ax^{3} + bx^{2} + cx + d} + 16b^{4}x^{4}e^{ax^{3} + bx^{2} + cx + d} + 32b^{3}cx^{3}e^{ax^{3} + bx^{2} + cx + d} + 24b^{2}c^{2}x^{2}e^{ax^{3} + bx^{2} + cx + d} + 8bc^{3}xe^{ax^{3} + bx^{2} + cx + d} + c^{4}e^{ax^{3} + bx^{2} + cx + d}\\ \end{split}\end{equation} \]

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