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。
\[ \begin{equation}\begin{split}【1/1】求函数{(\frac{1}{2})}^{e^{{2}^{x}}} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = {\frac{1}{2}}^{e^{{2}^{x}}}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( {\frac{1}{2}}^{e^{{2}^{x}}}\right)}{dx}\\=&({\frac{1}{2}}^{e^{{2}^{x}}}((e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})))ln(\frac{1}{2}) + \frac{(e^{{2}^{x}})(0)}{(\frac{1}{2})}))\\=&{2}^{x}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln(\frac{1}{2})ln(2)\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( {2}^{x}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln(\frac{1}{2})ln(2)\right)}{dx}\\=&({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})){\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln(\frac{1}{2})ln(2) + {2}^{x}({\frac{1}{2}}^{e^{{2}^{x}}}((e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})))ln(\frac{1}{2}) + \frac{(e^{{2}^{x}})(0)}{(\frac{1}{2})}))e^{{2}^{x}}ln(\frac{1}{2})ln(2) + {2}^{x}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)}))ln(\frac{1}{2})ln(2) + \frac{{2}^{x}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}*0ln(2)}{(\frac{1}{2})} + \frac{{2}^{x}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln(\frac{1}{2})*0}{(2)}\\=&{2}^{x}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{2}(2)ln(\frac{1}{2}) + {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{2}(\frac{1}{2})ln^{2}(2) + {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{2}(2)ln(\frac{1}{2})\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( {2}^{x}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{2}(2)ln(\frac{1}{2}) + {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{2}(\frac{1}{2})ln^{2}(2) + {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{2}(2)ln(\frac{1}{2})\right)}{dx}\\=&({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})){\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{2}(2)ln(\frac{1}{2}) + {2}^{x}({\frac{1}{2}}^{e^{{2}^{x}}}((e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})))ln(\frac{1}{2}) + \frac{(e^{{2}^{x}})(0)}{(\frac{1}{2})}))e^{{2}^{x}}ln^{2}(2)ln(\frac{1}{2}) + {2}^{x}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)}))ln^{2}(2)ln(\frac{1}{2}) + \frac{{2}^{x}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}*2ln(2)*0ln(\frac{1}{2})}{(2)} + \frac{{2}^{x}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{2}(2)*0}{(\frac{1}{2})} + ({2}^{(2x)}((2)ln(2) + \frac{(2x)(0)}{(2)})){\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{2}(\frac{1}{2})ln^{2}(2) + {2}^{(2x)}({\frac{1}{2}}^{e^{{2}^{x}}}((e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})))ln(\frac{1}{2}) + \frac{(e^{{2}^{x}})(0)}{(\frac{1}{2})}))e^{{{2}^{x}}*{2}}ln^{2}(\frac{1}{2})ln^{2}(2) + {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}*2e^{{2}^{x}}e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)}))ln^{2}(\frac{1}{2})ln^{2}(2) + \frac{{2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}*2ln(\frac{1}{2})*0ln^{2}(2)}{(\frac{1}{2})} + \frac{{2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{2}(\frac{1}{2})*2ln(2)*0}{(2)} + ({2}^{(2x)}((2)ln(2) + \frac{(2x)(0)}{(2)})){\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{2}(2)ln(\frac{1}{2}) + {2}^{(2x)}({\frac{1}{2}}^{e^{{2}^{x}}}((e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})))ln(\frac{1}{2}) + \frac{(e^{{2}^{x}})(0)}{(\frac{1}{2})}))e^{{2}^{x}}ln^{2}(2)ln(\frac{1}{2}) + {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)}))ln^{2}(2)ln(\frac{1}{2}) + \frac{{2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}*2ln(2)*0ln(\frac{1}{2})}{(2)} + \frac{{2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{2}(2)*0}{(\frac{1}{2})}\\=&{2}^{x}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{3}(2)ln(\frac{1}{2}) + {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{2}(\frac{1}{2})ln^{3}(2) + 3 * {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{3}(2)ln(\frac{1}{2}) + 2 * {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{3}(2)ln^{2}(\frac{1}{2}) + {2}^{(3x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{3}}ln^{3}(\frac{1}{2})ln^{3}(2) + 2 * {\frac{1}{2}}^{e^{{2}^{x}}}{2}^{(3x)}e^{{{2}^{x}}*{2}}ln^{3}(2)ln^{2}(\frac{1}{2}) + {2}^{(3x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{2}(\frac{1}{2})ln^{3}(2) + {2}^{(3x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{3}(2)ln(\frac{1}{2})\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( {2}^{x}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{3}(2)ln(\frac{1}{2}) + {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{2}(\frac{1}{2})ln^{3}(2) + 3 * {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{3}(2)ln(\frac{1}{2}) + 2 * {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{3}(2)ln^{2}(\frac{1}{2}) + {2}^{(3x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{3}}ln^{3}(\frac{1}{2})ln^{3}(2) + 2 * {\frac{1}{2}}^{e^{{2}^{x}}}{2}^{(3x)}e^{{{2}^{x}}*{2}}ln^{3}(2)ln^{2}(\frac{1}{2}) + {2}^{(3x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{2}(\frac{1}{2})ln^{3}(2) + {2}^{(3x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{3}(2)ln(\frac{1}{2})\right)}{dx}\\=&({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})){\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{3}(2)ln(\frac{1}{2}) + {2}^{x}({\frac{1}{2}}^{e^{{2}^{x}}}((e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})))ln(\frac{1}{2}) + \frac{(e^{{2}^{x}})(0)}{(\frac{1}{2})}))e^{{2}^{x}}ln^{3}(2)ln(\frac{1}{2}) + {2}^{x}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)}))ln^{3}(2)ln(\frac{1}{2}) + \frac{{2}^{x}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}*3ln^{2}(2)*0ln(\frac{1}{2})}{(2)} + \frac{{2}^{x}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{3}(2)*0}{(\frac{1}{2})} + ({2}^{(2x)}((2)ln(2) + \frac{(2x)(0)}{(2)})){\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{2}(\frac{1}{2})ln^{3}(2) + {2}^{(2x)}({\frac{1}{2}}^{e^{{2}^{x}}}((e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})))ln(\frac{1}{2}) + \frac{(e^{{2}^{x}})(0)}{(\frac{1}{2})}))e^{{{2}^{x}}*{2}}ln^{2}(\frac{1}{2})ln^{3}(2) + {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}*2e^{{2}^{x}}e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)}))ln^{2}(\frac{1}{2})ln^{3}(2) + \frac{{2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}*2ln(\frac{1}{2})*0ln^{3}(2)}{(\frac{1}{2})} + \frac{{2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{2}(\frac{1}{2})*3ln^{2}(2)*0}{(2)} + 3({2}^{(2x)}((2)ln(2) + \frac{(2x)(0)}{(2)})){\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{3}(2)ln(\frac{1}{2}) + 3 * {2}^{(2x)}({\frac{1}{2}}^{e^{{2}^{x}}}((e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})))ln(\frac{1}{2}) + \frac{(e^{{2}^{x}})(0)}{(\frac{1}{2})}))e^{{2}^{x}}ln^{3}(2)ln(\frac{1}{2}) + 3 * {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)}))ln^{3}(2)ln(\frac{1}{2}) + \frac{3 * {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}*3ln^{2}(2)*0ln(\frac{1}{2})}{(2)} + \frac{3 * {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{3}(2)*0}{(\frac{1}{2})} + 2({2}^{(2x)}((2)ln(2) + \frac{(2x)(0)}{(2)})){\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{3}(2)ln^{2}(\frac{1}{2}) + 2 * {2}^{(2x)}({\frac{1}{2}}^{e^{{2}^{x}}}((e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})))ln(\frac{1}{2}) + \frac{(e^{{2}^{x}})(0)}{(\frac{1}{2})}))e^{{{2}^{x}}*{2}}ln^{3}(2)ln^{2}(\frac{1}{2}) + 2 * {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}*2e^{{2}^{x}}e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)}))ln^{3}(2)ln^{2}(\frac{1}{2}) + \frac{2 * {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}*3ln^{2}(2)*0ln^{2}(\frac{1}{2})}{(2)} + \frac{2 * {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{3}(2)*2ln(\frac{1}{2})*0}{(\frac{1}{2})} + ({2}^{(3x)}((3)ln(2) + \frac{(3x)(0)}{(2)})){\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{3}}ln^{3}(\frac{1}{2})ln^{3}(2) + {2}^{(3x)}({\frac{1}{2}}^{e^{{2}^{x}}}((e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})))ln(\frac{1}{2}) + \frac{(e^{{2}^{x}})(0)}{(\frac{1}{2})}))e^{{{2}^{x}}*{3}}ln^{3}(\frac{1}{2})ln^{3}(2) + {2}^{(3x)}{\frac{1}{2}}^{e^{{2}^{x}}}*3e^{{{2}^{x}}*{2}}e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)}))ln^{3}(\frac{1}{2})ln^{3}(2) + \frac{{2}^{(3x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{3}}*3ln^{2}(\frac{1}{2})*0ln^{3}(2)}{(\frac{1}{2})} + \frac{{2}^{(3x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{3}}ln^{3}(\frac{1}{2})*3ln^{2}(2)*0}{(2)} + 2({\frac{1}{2}}^{e^{{2}^{x}}}((e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})))ln(\frac{1}{2}) + \frac{(e^{{2}^{x}})(0)}{(\frac{1}{2})})){2}^{(3x)}e^{{{2}^{x}}*{2}}ln^{3}(2)ln^{2}(\frac{1}{2}) + 2 * {\frac{1}{2}}^{e^{{2}^{x}}}({2}^{(3x)}((3)ln(2) + \frac{(3x)(0)}{(2)}))e^{{{2}^{x}}*{2}}ln^{3}(2)ln^{2}(\frac{1}{2}) + 2 * {\frac{1}{2}}^{e^{{2}^{x}}}{2}^{(3x)}*2e^{{2}^{x}}e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)}))ln^{3}(2)ln^{2}(\frac{1}{2}) + \frac{2 * {\frac{1}{2}}^{e^{{2}^{x}}}{2}^{(3x)}e^{{{2}^{x}}*{2}}*3ln^{2}(2)*0ln^{2}(\frac{1}{2})}{(2)} + \frac{2 * {\frac{1}{2}}^{e^{{2}^{x}}}{2}^{(3x)}e^{{{2}^{x}}*{2}}ln^{3}(2)*2ln(\frac{1}{2})*0}{(\frac{1}{2})} + ({2}^{(3x)}((3)ln(2) + \frac{(3x)(0)}{(2)})){\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{2}(\frac{1}{2})ln^{3}(2) + {2}^{(3x)}({\frac{1}{2}}^{e^{{2}^{x}}}((e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})))ln(\frac{1}{2}) + \frac{(e^{{2}^{x}})(0)}{(\frac{1}{2})}))e^{{{2}^{x}}*{2}}ln^{2}(\frac{1}{2})ln^{3}(2) + {2}^{(3x)}{\frac{1}{2}}^{e^{{2}^{x}}}*2e^{{2}^{x}}e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)}))ln^{2}(\frac{1}{2})ln^{3}(2) + \frac{{2}^{(3x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}*2ln(\frac{1}{2})*0ln^{3}(2)}{(\frac{1}{2})} + \frac{{2}^{(3x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{2}(\frac{1}{2})*3ln^{2}(2)*0}{(2)} + ({2}^{(3x)}((3)ln(2) + \frac{(3x)(0)}{(2)})){\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{3}(2)ln(\frac{1}{2}) + {2}^{(3x)}({\frac{1}{2}}^{e^{{2}^{x}}}((e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)})))ln(\frac{1}{2}) + \frac{(e^{{2}^{x}})(0)}{(\frac{1}{2})}))e^{{2}^{x}}ln^{3}(2)ln(\frac{1}{2}) + {2}^{(3x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}({2}^{x}((1)ln(2) + \frac{(x)(0)}{(2)}))ln^{3}(2)ln(\frac{1}{2}) + \frac{{2}^{(3x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}*3ln^{2}(2)*0ln(\frac{1}{2})}{(2)} + \frac{{2}^{(3x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{3}(2)*0}{(\frac{1}{2})}\\=&{2}^{x}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{4}(2)ln(\frac{1}{2}) + {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{2}(\frac{1}{2})ln^{4}(2) + 7 * {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{4}(2)ln(\frac{1}{2}) + 6 * {2}^{(2x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{4}(2)ln^{2}(\frac{1}{2}) + 3 * {2}^{(3x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{3}}ln^{3}(\frac{1}{2})ln^{4}(2) + 6 * {\frac{1}{2}}^{e^{{2}^{x}}}{2}^{(3x)}e^{{{2}^{x}}*{2}}ln^{4}(2)ln^{2}(\frac{1}{2}) + 3 * {2}^{(3x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{2}(\frac{1}{2})ln^{4}(2) + 6 * {2}^{(3x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{4}(2)ln(\frac{1}{2}) + 3 * {2}^{(3x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{3}}ln^{4}(2)ln^{3}(\frac{1}{2}) + {2}^{(4x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{4}}ln^{4}(\frac{1}{2})ln^{4}(2) + 3 * {\frac{1}{2}}^{e^{{2}^{x}}}{2}^{(4x)}e^{{{2}^{x}}*{3}}ln^{4}(2)ln^{3}(\frac{1}{2}) + 3 * {2}^{(4x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{3}}ln^{3}(\frac{1}{2})ln^{4}(2) + 9 * {2}^{(3x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{4}(2)ln^{2}(\frac{1}{2}) + 4 * {2}^{(4x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{4}(2)ln^{2}(\frac{1}{2}) + 2 * {\frac{1}{2}}^{e^{{2}^{x}}}{2}^{(4x)}e^{{{2}^{x}}*{2}}ln^{4}(2)ln^{2}(\frac{1}{2}) + {2}^{(4x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{{2}^{x}}*{2}}ln^{2}(\frac{1}{2})ln^{4}(2) + {2}^{(4x)}{\frac{1}{2}}^{e^{{2}^{x}}}e^{{2}^{x}}ln^{4}(2)ln(\frac{1}{2})\\ \end{split}\end{equation} \]

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