求导函数：
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
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。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{({a}^{x} - b - cx){\frac{1}{x}}^{e}}{d} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{{a}^{x}{\frac{1}{x}}^{e}}{d} - \frac{b{\frac{1}{x}}^{e}}{d} - \frac{cx{\frac{1}{x}}^{e}}{d}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{{a}^{x}{\frac{1}{x}}^{e}}{d} - \frac{b{\frac{1}{x}}^{e}}{d} - \frac{cx{\frac{1}{x}}^{e}}{d}\right)}{dx}\\=&\frac{({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){\frac{1}{x}}^{e}}{d} + \frac{{a}^{x}({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))}{d} - \frac{b({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))}{d} - \frac{c{\frac{1}{x}}^{e}}{d} - \frac{cx({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))}{d}\\=&\frac{{a}^{x}{\frac{1}{x}}^{e}ln(a)}{d} - \frac{{\frac{1}{x}}^{e}{a}^{x}e}{dx} + \frac{b{\frac{1}{x}}^{e}e}{dx} + \frac{c{\frac{1}{x}}^{e}e}{d} - \frac{c{\frac{1}{x}}^{e}}{d}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{{a}^{x}{\frac{1}{x}}^{e}ln(a)}{d} - \frac{{\frac{1}{x}}^{e}{a}^{x}e}{dx} + \frac{b{\frac{1}{x}}^{e}e}{dx} + \frac{c{\frac{1}{x}}^{e}e}{d} - \frac{c{\frac{1}{x}}^{e}}{d}\right)}{dx}\\=&\frac{({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){\frac{1}{x}}^{e}ln(a)}{d} + \frac{{a}^{x}({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))ln(a)}{d} + \frac{{a}^{x}{\frac{1}{x}}^{e}*0}{d(a)} - \frac{-{\frac{1}{x}}^{e}{a}^{x}e}{dx^{2}} - \frac{({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){a}^{x}e}{dx} - \frac{{\frac{1}{x}}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e}{dx} - \frac{{\frac{1}{x}}^{e}{a}^{x}*0}{dx} + \frac{b*-{\frac{1}{x}}^{e}e}{dx^{2}} + \frac{b({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e}{dx} + \frac{b{\frac{1}{x}}^{e}*0}{dx} + \frac{c({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e}{d} + \frac{c{\frac{1}{x}}^{e}*0}{d} - \frac{c({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))}{d}\\=&\frac{{a}^{x}{\frac{1}{x}}^{e}ln^{2}(a)}{d} - \frac{{\frac{1}{x}}^{e}{a}^{x}eln(a)}{dx} - \frac{{a}^{x}{\frac{1}{x}}^{e}eln(a)}{dx} + \frac{{\frac{1}{x}}^{e}{a}^{x}e^{2}}{dx^{2}} + \frac{{\frac{1}{x}}^{e}{a}^{x}e}{dx^{2}} - \frac{b{\frac{1}{x}}^{e}e}{dx^{2}} - \frac{b{\frac{1}{x}}^{e}e^{2}}{dx^{2}} - \frac{c{\frac{1}{x}}^{e}e^{2}}{dx} + \frac{c{\frac{1}{x}}^{e}e}{dx}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{{a}^{x}{\frac{1}{x}}^{e}ln^{2}(a)}{d} - \frac{{\frac{1}{x}}^{e}{a}^{x}eln(a)}{dx} - \frac{{a}^{x}{\frac{1}{x}}^{e}eln(a)}{dx} + \frac{{\frac{1}{x}}^{e}{a}^{x}e^{2}}{dx^{2}} + \frac{{\frac{1}{x}}^{e}{a}^{x}e}{dx^{2}} - \frac{b{\frac{1}{x}}^{e}e}{dx^{2}} - \frac{b{\frac{1}{x}}^{e}e^{2}}{dx^{2}} - \frac{c{\frac{1}{x}}^{e}e^{2}}{dx} + \frac{c{\frac{1}{x}}^{e}e}{dx}\right)}{dx}\\=&\frac{({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){\frac{1}{x}}^{e}ln^{2}(a)}{d} + \frac{{a}^{x}({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))ln^{2}(a)}{d} + \frac{{a}^{x}{\frac{1}{x}}^{e}*2ln(a)*0}{d(a)} - \frac{-{\frac{1}{x}}^{e}{a}^{x}eln(a)}{dx^{2}} - \frac{({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){a}^{x}eln(a)}{dx} - \frac{{\frac{1}{x}}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))eln(a)}{dx} - \frac{{\frac{1}{x}}^{e}{a}^{x}*0ln(a)}{dx} - \frac{{\frac{1}{x}}^{e}{a}^{x}e*0}{dx(a)} - \frac{-{a}^{x}{\frac{1}{x}}^{e}eln(a)}{dx^{2}} - \frac{({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){\frac{1}{x}}^{e}eln(a)}{dx} - \frac{{a}^{x}({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))eln(a)}{dx} - \frac{{a}^{x}{\frac{1}{x}}^{e}*0ln(a)}{dx} - \frac{{a}^{x}{\frac{1}{x}}^{e}e*0}{dx(a)} + \frac{-2{\frac{1}{x}}^{e}{a}^{x}e^{2}}{dx^{3}} + \frac{({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){a}^{x}e^{2}}{dx^{2}} + \frac{{\frac{1}{x}}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e^{2}}{dx^{2}} + \frac{{\frac{1}{x}}^{e}{a}^{x}*2e*0}{dx^{2}} + \frac{-2{\frac{1}{x}}^{e}{a}^{x}e}{dx^{3}} + \frac{({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){a}^{x}e}{dx^{2}} + \frac{{\frac{1}{x}}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e}{dx^{2}} + \frac{{\frac{1}{x}}^{e}{a}^{x}*0}{dx^{2}} - \frac{b*-2{\frac{1}{x}}^{e}e}{dx^{3}} - \frac{b({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e}{dx^{2}} - \frac{b{\frac{1}{x}}^{e}*0}{dx^{2}} - \frac{b*-2{\frac{1}{x}}^{e}e^{2}}{dx^{3}} - \frac{b({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e^{2}}{dx^{2}} - \frac{b{\frac{1}{x}}^{e}*2e*0}{dx^{2}} - \frac{c*-{\frac{1}{x}}^{e}e^{2}}{dx^{2}} - \frac{c({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e^{2}}{dx} - \frac{c{\frac{1}{x}}^{e}*2e*0}{dx} + \frac{c*-{\frac{1}{x}}^{e}e}{dx^{2}} + \frac{c({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e}{dx} + \frac{c{\frac{1}{x}}^{e}*0}{dx}\\=&\frac{{a}^{x}{\frac{1}{x}}^{e}ln^{3}(a)}{d} + \frac{{\frac{1}{x}}^{e}{a}^{x}eln(a)}{dx^{2}} - \frac{2{a}^{x}{\frac{1}{x}}^{e}eln^{2}(a)}{dx} + \frac{2{\frac{1}{x}}^{e}{a}^{x}e^{2}ln(a)}{dx^{2}} - \frac{{\frac{1}{x}}^{e}{a}^{x}eln^{2}(a)}{dx} + \frac{2{a}^{x}{\frac{1}{x}}^{e}eln(a)}{dx^{2}} + \frac{{a}^{x}{\frac{1}{x}}^{e}e^{2}ln(a)}{dx^{2}} - \frac{{\frac{1}{x}}^{e}{a}^{x}e^{3}}{dx^{3}} - \frac{3{\frac{1}{x}}^{e}{a}^{x}e^{2}}{dx^{3}} - \frac{2{\frac{1}{x}}^{e}{a}^{x}e}{dx^{3}} + \frac{2b{\frac{1}{x}}^{e}e}{dx^{3}} + \frac{3b{\frac{1}{x}}^{e}e^{2}}{dx^{3}} + \frac{b{\frac{1}{x}}^{e}e^{3}}{dx^{3}} + \frac{c{\frac{1}{x}}^{e}e^{3}}{dx^{2}} - \frac{c{\frac{1}{x}}^{e}e}{dx^{2}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{{a}^{x}{\frac{1}{x}}^{e}ln^{3}(a)}{d} + \frac{{\frac{1}{x}}^{e}{a}^{x}eln(a)}{dx^{2}} - \frac{2{a}^{x}{\frac{1}{x}}^{e}eln^{2}(a)}{dx} + \frac{2{\frac{1}{x}}^{e}{a}^{x}e^{2}ln(a)}{dx^{2}} - \frac{{\frac{1}{x}}^{e}{a}^{x}eln^{2}(a)}{dx} + \frac{2{a}^{x}{\frac{1}{x}}^{e}eln(a)}{dx^{2}} + \frac{{a}^{x}{\frac{1}{x}}^{e}e^{2}ln(a)}{dx^{2}} - \frac{{\frac{1}{x}}^{e}{a}^{x}e^{3}}{dx^{3}} - \frac{3{\frac{1}{x}}^{e}{a}^{x}e^{2}}{dx^{3}} - \frac{2{\frac{1}{x}}^{e}{a}^{x}e}{dx^{3}} + \frac{2b{\frac{1}{x}}^{e}e}{dx^{3}} + \frac{3b{\frac{1}{x}}^{e}e^{2}}{dx^{3}} + \frac{b{\frac{1}{x}}^{e}e^{3}}{dx^{3}} + \frac{c{\frac{1}{x}}^{e}e^{3}}{dx^{2}} - \frac{c{\frac{1}{x}}^{e}e}{dx^{2}}\right)}{dx}\\=&\frac{({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){\frac{1}{x}}^{e}ln^{3}(a)}{d} + \frac{{a}^{x}({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))ln^{3}(a)}{d} + \frac{{a}^{x}{\frac{1}{x}}^{e}*3ln^{2}(a)*0}{d(a)} + \frac{-2{\frac{1}{x}}^{e}{a}^{x}eln(a)}{dx^{3}} + \frac{({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){a}^{x}eln(a)}{dx^{2}} + \frac{{\frac{1}{x}}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))eln(a)}{dx^{2}} + \frac{{\frac{1}{x}}^{e}{a}^{x}*0ln(a)}{dx^{2}} + \frac{{\frac{1}{x}}^{e}{a}^{x}e*0}{dx^{2}(a)} - \frac{2*-{a}^{x}{\frac{1}{x}}^{e}eln^{2}(a)}{dx^{2}} - \frac{2({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){\frac{1}{x}}^{e}eln^{2}(a)}{dx} - \frac{2{a}^{x}({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))eln^{2}(a)}{dx} - \frac{2{a}^{x}{\frac{1}{x}}^{e}*0ln^{2}(a)}{dx} - \frac{2{a}^{x}{\frac{1}{x}}^{e}e*2ln(a)*0}{dx(a)} + \frac{2*-2{\frac{1}{x}}^{e}{a}^{x}e^{2}ln(a)}{dx^{3}} + \frac{2({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){a}^{x}e^{2}ln(a)}{dx^{2}} + \frac{2{\frac{1}{x}}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e^{2}ln(a)}{dx^{2}} + \frac{2{\frac{1}{x}}^{e}{a}^{x}*2e*0ln(a)}{dx^{2}} + \frac{2{\frac{1}{x}}^{e}{a}^{x}e^{2}*0}{dx^{2}(a)} - \frac{-{\frac{1}{x}}^{e}{a}^{x}eln^{2}(a)}{dx^{2}} - \frac{({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){a}^{x}eln^{2}(a)}{dx} - \frac{{\frac{1}{x}}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))eln^{2}(a)}{dx} - \frac{{\frac{1}{x}}^{e}{a}^{x}*0ln^{2}(a)}{dx} - \frac{{\frac{1}{x}}^{e}{a}^{x}e*2ln(a)*0}{dx(a)} + \frac{2*-2{a}^{x}{\frac{1}{x}}^{e}eln(a)}{dx^{3}} + \frac{2({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){\frac{1}{x}}^{e}eln(a)}{dx^{2}} + \frac{2{a}^{x}({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))eln(a)}{dx^{2}} + \frac{2{a}^{x}{\frac{1}{x}}^{e}*0ln(a)}{dx^{2}} + \frac{2{a}^{x}{\frac{1}{x}}^{e}e*0}{dx^{2}(a)} + \frac{-2{a}^{x}{\frac{1}{x}}^{e}e^{2}ln(a)}{dx^{3}} + \frac{({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){\frac{1}{x}}^{e}e^{2}ln(a)}{dx^{2}} + \frac{{a}^{x}({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e^{2}ln(a)}{dx^{2}} + \frac{{a}^{x}{\frac{1}{x}}^{e}*2e*0ln(a)}{dx^{2}} + \frac{{a}^{x}{\frac{1}{x}}^{e}e^{2}*0}{dx^{2}(a)} - \frac{-3{\frac{1}{x}}^{e}{a}^{x}e^{3}}{dx^{4}} - \frac{({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){a}^{x}e^{3}}{dx^{3}} - \frac{{\frac{1}{x}}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e^{3}}{dx^{3}} - \frac{{\frac{1}{x}}^{e}{a}^{x}*3e^{2}*0}{dx^{3}} - \frac{3*-3{\frac{1}{x}}^{e}{a}^{x}e^{2}}{dx^{4}} - \frac{3({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){a}^{x}e^{2}}{dx^{3}} - \frac{3{\frac{1}{x}}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e^{2}}{dx^{3}} - \frac{3{\frac{1}{x}}^{e}{a}^{x}*2e*0}{dx^{3}} - \frac{2*-3{\frac{1}{x}}^{e}{a}^{x}e}{dx^{4}} - \frac{2({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){a}^{x}e}{dx^{3}} - \frac{2{\frac{1}{x}}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e}{dx^{3}} - \frac{2{\frac{1}{x}}^{e}{a}^{x}*0}{dx^{3}} + \frac{2b*-3{\frac{1}{x}}^{e}e}{dx^{4}} + \frac{2b({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e}{dx^{3}} + \frac{2b{\frac{1}{x}}^{e}*0}{dx^{3}} + \frac{3b*-3{\frac{1}{x}}^{e}e^{2}}{dx^{4}} + \frac{3b({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e^{2}}{dx^{3}} + \frac{3b{\frac{1}{x}}^{e}*2e*0}{dx^{3}} + \frac{b*-3{\frac{1}{x}}^{e}e^{3}}{dx^{4}} + \frac{b({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e^{3}}{dx^{3}} + \frac{b{\frac{1}{x}}^{e}*3e^{2}*0}{dx^{3}} + \frac{c*-2{\frac{1}{x}}^{e}e^{3}}{dx^{3}} + \frac{c({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e^{3}}{dx^{2}} + \frac{c{\frac{1}{x}}^{e}*3e^{2}*0}{dx^{2}} - \frac{c*-2{\frac{1}{x}}^{e}e}{dx^{3}} - \frac{c({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e}{dx^{2}} - \frac{c{\frac{1}{x}}^{e}*0}{dx^{2}}\\=&\frac{{a}^{x}{\frac{1}{x}}^{e}ln^{4}(a)}{d} - \frac{2{\frac{1}{x}}^{e}{a}^{x}eln(a)}{dx^{3}} + \frac{5{a}^{x}{\frac{1}{x}}^{e}eln^{2}(a)}{dx^{2}} - \frac{7{\frac{1}{x}}^{e}{a}^{x}e^{2}ln(a)}{dx^{3}} - \frac{3{a}^{x}{\frac{1}{x}}^{e}eln^{3}(a)}{dx} + \frac{3{\frac{1}{x}}^{e}{a}^{x}e^{2}ln^{2}(a)}{dx^{2}} + \frac{3{a}^{x}{\frac{1}{x}}^{e}e^{2}ln^{2}(a)}{dx^{2}} - \frac{3{\frac{1}{x}}^{e}{a}^{x}e^{3}ln(a)}{dx^{3}} - \frac{{\frac{1}{x}}^{e}{a}^{x}eln^{3}(a)}{dx} + \frac{{\frac{1}{x}}^{e}{a}^{x}eln^{2}(a)}{dx^{2}} - \frac{6{a}^{x}{\frac{1}{x}}^{e}eln(a)}{dx^{3}} - \frac{5{a}^{x}{\frac{1}{x}}^{e}e^{2}ln(a)}{dx^{3}} - \frac{{a}^{x}{\frac{1}{x}}^{e}e^{3}ln(a)}{dx^{3}} + \frac{{\frac{1}{x}}^{e}{a}^{x}e^{4}}{dx^{4}} + \frac{6{\frac{1}{x}}^{e}{a}^{x}e^{3}}{dx^{4}} + \frac{11{\frac{1}{x}}^{e}{a}^{x}e^{2}}{dx^{4}} + \frac{6{\frac{1}{x}}^{e}{a}^{x}e}{dx^{4}} - \frac{6b{\frac{1}{x}}^{e}e}{dx^{4}} - \frac{11b{\frac{1}{x}}^{e}e^{2}}{dx^{4}} - \frac{6b{\frac{1}{x}}^{e}e^{3}}{dx^{4}} - \frac{b{\frac{1}{x}}^{e}e^{4}}{dx^{4}} - \frac{2c{\frac{1}{x}}^{e}e^{3}}{dx^{3}} - \frac{c{\frac{1}{x}}^{e}e^{4}}{dx^{3}} + \frac{2c{\frac{1}{x}}^{e}e}{dx^{3}} + \frac{c{\frac{1}{x}}^{e}e^{2}}{dx^{3}}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{({a}^{x} - b - cx){\frac{1}{x}}^{e}}{d} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{{a}^{x}{\frac{1}{x}}^{e}}{d} - \frac{b{\frac{1}{x}}^{e}}{d} - \frac{cx{\frac{1}{x}}^{e}}{d}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{{a}^{x}{\frac{1}{x}}^{e}}{d} - \frac{b{\frac{1}{x}}^{e}}{d} - \frac{cx{\frac{1}{x}}^{e}}{d}\right)}{dx}\\=&\frac{({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){\frac{1}{x}}^{e}}{d} + \frac{{a}^{x}({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))}{d} - \frac{b({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))}{d} - \frac{c{\frac{1}{x}}^{e}}{d} - \frac{cx({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))}{d}\\=&\frac{{a}^{x}{\frac{1}{x}}^{e}ln(a)}{d} - \frac{{\frac{1}{x}}^{e}{a}^{x}e}{dx} + \frac{b{\frac{1}{x}}^{e}e}{dx} + \frac{c{\frac{1}{x}}^{e}e}{d} - \frac{c{\frac{1}{x}}^{e}}{d}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{{a}^{x}{\frac{1}{x}}^{e}ln(a)}{d} - \frac{{\frac{1}{x}}^{e}{a}^{x}e}{dx} + \frac{b{\frac{1}{x}}^{e}e}{dx} + \frac{c{\frac{1}{x}}^{e}e}{d} - \frac{c{\frac{1}{x}}^{e}}{d}\right)}{dx}\\=&\frac{({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){\frac{1}{x}}^{e}ln(a)}{d} + \frac{{a}^{x}({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))ln(a)}{d} + \frac{{a}^{x}{\frac{1}{x}}^{e}*0}{d(a)} - \frac{-{\frac{1}{x}}^{e}{a}^{x}e}{dx^{2}} - \frac{({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){a}^{x}e}{dx} - \frac{{\frac{1}{x}}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e}{dx} - \frac{{\frac{1}{x}}^{e}{a}^{x}*0}{dx} + \frac{b*-{\frac{1}{x}}^{e}e}{dx^{2}} + \frac{b({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e}{dx} + \frac{b{\frac{1}{x}}^{e}*0}{dx} + \frac{c({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e}{d} + \frac{c{\frac{1}{x}}^{e}*0}{d} - \frac{c({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))}{d}\\=&\frac{{a}^{x}{\frac{1}{x}}^{e}ln^{2}(a)}{d} - \frac{{\frac{1}{x}}^{e}{a}^{x}eln(a)}{dx} - \frac{{a}^{x}{\frac{1}{x}}^{e}eln(a)}{dx} + \frac{{\frac{1}{x}}^{e}{a}^{x}e^{2}}{dx^{2}} + \frac{{\frac{1}{x}}^{e}{a}^{x}e}{dx^{2}} - \frac{b{\frac{1}{x}}^{e}e}{dx^{2}} - \frac{b{\frac{1}{x}}^{e}e^{2}}{dx^{2}} - \frac{c{\frac{1}{x}}^{e}e^{2}}{dx} + \frac{c{\frac{1}{x}}^{e}e}{dx}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{{a}^{x}{\frac{1}{x}}^{e}ln^{2}(a)}{d} - \frac{{\frac{1}{x}}^{e}{a}^{x}eln(a)}{dx} - \frac{{a}^{x}{\frac{1}{x}}^{e}eln(a)}{dx} + \frac{{\frac{1}{x}}^{e}{a}^{x}e^{2}}{dx^{2}} + \frac{{\frac{1}{x}}^{e}{a}^{x}e}{dx^{2}} - \frac{b{\frac{1}{x}}^{e}e}{dx^{2}} - \frac{b{\frac{1}{x}}^{e}e^{2}}{dx^{2}} - \frac{c{\frac{1}{x}}^{e}e^{2}}{dx} + \frac{c{\frac{1}{x}}^{e}e}{dx}\right)}{dx}\\=&\frac{({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){\frac{1}{x}}^{e}ln^{2}(a)}{d} + \frac{{a}^{x}({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))ln^{2}(a)}{d} + \frac{{a}^{x}{\frac{1}{x}}^{e}*2ln(a)*0}{d(a)} - \frac{-{\frac{1}{x}}^{e}{a}^{x}eln(a)}{dx^{2}} - \frac{({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){a}^{x}eln(a)}{dx} - \frac{{\frac{1}{x}}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))eln(a)}{dx} - \frac{{\frac{1}{x}}^{e}{a}^{x}*0ln(a)}{dx} - \frac{{\frac{1}{x}}^{e}{a}^{x}e*0}{dx(a)} - \frac{-{a}^{x}{\frac{1}{x}}^{e}eln(a)}{dx^{2}} - \frac{({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){\frac{1}{x}}^{e}eln(a)}{dx} - \frac{{a}^{x}({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))eln(a)}{dx} - \frac{{a}^{x}{\frac{1}{x}}^{e}*0ln(a)}{dx} - \frac{{a}^{x}{\frac{1}{x}}^{e}e*0}{dx(a)} + \frac{-2{\frac{1}{x}}^{e}{a}^{x}e^{2}}{dx^{3}} + \frac{({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){a}^{x}e^{2}}{dx^{2}} + \frac{{\frac{1}{x}}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e^{2}}{dx^{2}} + \frac{{\frac{1}{x}}^{e}{a}^{x}*2e*0}{dx^{2}} + \frac{-2{\frac{1}{x}}^{e}{a}^{x}e}{dx^{3}} + \frac{({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){a}^{x}e}{dx^{2}} + \frac{{\frac{1}{x}}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e}{dx^{2}} + \frac{{\frac{1}{x}}^{e}{a}^{x}*0}{dx^{2}} - \frac{b*-2{\frac{1}{x}}^{e}e}{dx^{3}} - \frac{b({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e}{dx^{2}} - \frac{b{\frac{1}{x}}^{e}*0}{dx^{2}} - \frac{b*-2{\frac{1}{x}}^{e}e^{2}}{dx^{3}} - \frac{b({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e^{2}}{dx^{2}} - \frac{b{\frac{1}{x}}^{e}*2e*0}{dx^{2}} - \frac{c*-{\frac{1}{x}}^{e}e^{2}}{dx^{2}} - \frac{c({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e^{2}}{dx} - \frac{c{\frac{1}{x}}^{e}*2e*0}{dx} + \frac{c*-{\frac{1}{x}}^{e}e}{dx^{2}} + \frac{c({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e}{dx} + \frac{c{\frac{1}{x}}^{e}*0}{dx}\\=&\frac{{a}^{x}{\frac{1}{x}}^{e}ln^{3}(a)}{d} + \frac{{\frac{1}{x}}^{e}{a}^{x}eln(a)}{dx^{2}} - \frac{2{a}^{x}{\frac{1}{x}}^{e}eln^{2}(a)}{dx} + \frac{2{\frac{1}{x}}^{e}{a}^{x}e^{2}ln(a)}{dx^{2}} - \frac{{\frac{1}{x}}^{e}{a}^{x}eln^{2}(a)}{dx} + \frac{2{a}^{x}{\frac{1}{x}}^{e}eln(a)}{dx^{2}} + \frac{{a}^{x}{\frac{1}{x}}^{e}e^{2}ln(a)}{dx^{2}} - \frac{{\frac{1}{x}}^{e}{a}^{x}e^{3}}{dx^{3}} - \frac{3{\frac{1}{x}}^{e}{a}^{x}e^{2}}{dx^{3}} - \frac{2{\frac{1}{x}}^{e}{a}^{x}e}{dx^{3}} + \frac{2b{\frac{1}{x}}^{e}e}{dx^{3}} + \frac{3b{\frac{1}{x}}^{e}e^{2}}{dx^{3}} + \frac{b{\frac{1}{x}}^{e}e^{3}}{dx^{3}} + \frac{c{\frac{1}{x}}^{e}e^{3}}{dx^{2}} - \frac{c{\frac{1}{x}}^{e}e}{dx^{2}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{{a}^{x}{\frac{1}{x}}^{e}ln^{3}(a)}{d} + \frac{{\frac{1}{x}}^{e}{a}^{x}eln(a)}{dx^{2}} - \frac{2{a}^{x}{\frac{1}{x}}^{e}eln^{2}(a)}{dx} + \frac{2{\frac{1}{x}}^{e}{a}^{x}e^{2}ln(a)}{dx^{2}} - \frac{{\frac{1}{x}}^{e}{a}^{x}eln^{2}(a)}{dx} + \frac{2{a}^{x}{\frac{1}{x}}^{e}eln(a)}{dx^{2}} + \frac{{a}^{x}{\frac{1}{x}}^{e}e^{2}ln(a)}{dx^{2}} - \frac{{\frac{1}{x}}^{e}{a}^{x}e^{3}}{dx^{3}} - \frac{3{\frac{1}{x}}^{e}{a}^{x}e^{2}}{dx^{3}} - \frac{2{\frac{1}{x}}^{e}{a}^{x}e}{dx^{3}} + \frac{2b{\frac{1}{x}}^{e}e}{dx^{3}} + \frac{3b{\frac{1}{x}}^{e}e^{2}}{dx^{3}} + \frac{b{\frac{1}{x}}^{e}e^{3}}{dx^{3}} + \frac{c{\frac{1}{x}}^{e}e^{3}}{dx^{2}} - \frac{c{\frac{1}{x}}^{e}e}{dx^{2}}\right)}{dx}\\=&\frac{({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){\frac{1}{x}}^{e}ln^{3}(a)}{d} + \frac{{a}^{x}({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))ln^{3}(a)}{d} + \frac{{a}^{x}{\frac{1}{x}}^{e}*3ln^{2}(a)*0}{d(a)} + \frac{-2{\frac{1}{x}}^{e}{a}^{x}eln(a)}{dx^{3}} + \frac{({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){a}^{x}eln(a)}{dx^{2}} + \frac{{\frac{1}{x}}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))eln(a)}{dx^{2}} + \frac{{\frac{1}{x}}^{e}{a}^{x}*0ln(a)}{dx^{2}} + \frac{{\frac{1}{x}}^{e}{a}^{x}e*0}{dx^{2}(a)} - \frac{2*-{a}^{x}{\frac{1}{x}}^{e}eln^{2}(a)}{dx^{2}} - \frac{2({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){\frac{1}{x}}^{e}eln^{2}(a)}{dx} - \frac{2{a}^{x}({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))eln^{2}(a)}{dx} - \frac{2{a}^{x}{\frac{1}{x}}^{e}*0ln^{2}(a)}{dx} - \frac{2{a}^{x}{\frac{1}{x}}^{e}e*2ln(a)*0}{dx(a)} + \frac{2*-2{\frac{1}{x}}^{e}{a}^{x}e^{2}ln(a)}{dx^{3}} + \frac{2({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){a}^{x}e^{2}ln(a)}{dx^{2}} + \frac{2{\frac{1}{x}}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e^{2}ln(a)}{dx^{2}} + \frac{2{\frac{1}{x}}^{e}{a}^{x}*2e*0ln(a)}{dx^{2}} + \frac{2{\frac{1}{x}}^{e}{a}^{x}e^{2}*0}{dx^{2}(a)} - \frac{-{\frac{1}{x}}^{e}{a}^{x}eln^{2}(a)}{dx^{2}} - \frac{({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){a}^{x}eln^{2}(a)}{dx} - \frac{{\frac{1}{x}}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))eln^{2}(a)}{dx} - \frac{{\frac{1}{x}}^{e}{a}^{x}*0ln^{2}(a)}{dx} - \frac{{\frac{1}{x}}^{e}{a}^{x}e*2ln(a)*0}{dx(a)} + \frac{2*-2{a}^{x}{\frac{1}{x}}^{e}eln(a)}{dx^{3}} + \frac{2({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){\frac{1}{x}}^{e}eln(a)}{dx^{2}} + \frac{2{a}^{x}({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))eln(a)}{dx^{2}} + \frac{2{a}^{x}{\frac{1}{x}}^{e}*0ln(a)}{dx^{2}} + \frac{2{a}^{x}{\frac{1}{x}}^{e}e*0}{dx^{2}(a)} + \frac{-2{a}^{x}{\frac{1}{x}}^{e}e^{2}ln(a)}{dx^{3}} + \frac{({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){\frac{1}{x}}^{e}e^{2}ln(a)}{dx^{2}} + \frac{{a}^{x}({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e^{2}ln(a)}{dx^{2}} + \frac{{a}^{x}{\frac{1}{x}}^{e}*2e*0ln(a)}{dx^{2}} + \frac{{a}^{x}{\frac{1}{x}}^{e}e^{2}*0}{dx^{2}(a)} - \frac{-3{\frac{1}{x}}^{e}{a}^{x}e^{3}}{dx^{4}} - \frac{({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){a}^{x}e^{3}}{dx^{3}} - \frac{{\frac{1}{x}}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e^{3}}{dx^{3}} - \frac{{\frac{1}{x}}^{e}{a}^{x}*3e^{2}*0}{dx^{3}} - \frac{3*-3{\frac{1}{x}}^{e}{a}^{x}e^{2}}{dx^{4}} - \frac{3({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){a}^{x}e^{2}}{dx^{3}} - \frac{3{\frac{1}{x}}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e^{2}}{dx^{3}} - \frac{3{\frac{1}{x}}^{e}{a}^{x}*2e*0}{dx^{3}} - \frac{2*-3{\frac{1}{x}}^{e}{a}^{x}e}{dx^{4}} - \frac{2({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})})){a}^{x}e}{dx^{3}} - \frac{2{\frac{1}{x}}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e}{dx^{3}} - \frac{2{\frac{1}{x}}^{e}{a}^{x}*0}{dx^{3}} + \frac{2b*-3{\frac{1}{x}}^{e}e}{dx^{4}} + \frac{2b({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e}{dx^{3}} + \frac{2b{\frac{1}{x}}^{e}*0}{dx^{3}} + \frac{3b*-3{\frac{1}{x}}^{e}e^{2}}{dx^{4}} + \frac{3b({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e^{2}}{dx^{3}} + \frac{3b{\frac{1}{x}}^{e}*2e*0}{dx^{3}} + \frac{b*-3{\frac{1}{x}}^{e}e^{3}}{dx^{4}} + \frac{b({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e^{3}}{dx^{3}} + \frac{b{\frac{1}{x}}^{e}*3e^{2}*0}{dx^{3}} + \frac{c*-2{\frac{1}{x}}^{e}e^{3}}{dx^{3}} + \frac{c({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e^{3}}{dx^{2}} + \frac{c{\frac{1}{x}}^{e}*3e^{2}*0}{dx^{2}} - \frac{c*-2{\frac{1}{x}}^{e}e}{dx^{3}} - \frac{c({\frac{1}{x}}^{e}((0)ln(\frac{1}{x}) + \frac{(e)(\frac{-1}{x^{2}})}{(\frac{1}{x})}))e}{dx^{2}} - \frac{c{\frac{1}{x}}^{e}*0}{dx^{2}}\\=&\frac{{a}^{x}{\frac{1}{x}}^{e}ln^{4}(a)}{d} - \frac{2{\frac{1}{x}}^{e}{a}^{x}eln(a)}{dx^{3}} + \frac{5{a}^{x}{\frac{1}{x}}^{e}eln^{2}(a)}{dx^{2}} - \frac{7{\frac{1}{x}}^{e}{a}^{x}e^{2}ln(a)}{dx^{3}} - \frac{3{a}^{x}{\frac{1}{x}}^{e}eln^{3}(a)}{dx} + \frac{3{\frac{1}{x}}^{e}{a}^{x}e^{2}ln^{2}(a)}{dx^{2}} + \frac{3{a}^{x}{\frac{1}{x}}^{e}e^{2}ln^{2}(a)}{dx^{2}} - \frac{3{\frac{1}{x}}^{e}{a}^{x}e^{3}ln(a)}{dx^{3}} - \frac{{\frac{1}{x}}^{e}{a}^{x}eln^{3}(a)}{dx} + \frac{{\frac{1}{x}}^{e}{a}^{x}eln^{2}(a)}{dx^{2}} - \frac{6{a}^{x}{\frac{1}{x}}^{e}eln(a)}{dx^{3}} - \frac{5{a}^{x}{\frac{1}{x}}^{e}e^{2}ln(a)}{dx^{3}} - \frac{{a}^{x}{\frac{1}{x}}^{e}e^{3}ln(a)}{dx^{3}} + \frac{{\frac{1}{x}}^{e}{a}^{x}e^{4}}{dx^{4}} + \frac{6{\frac{1}{x}}^{e}{a}^{x}e^{3}}{dx^{4}} + \frac{11{\frac{1}{x}}^{e}{a}^{x}e^{2}}{dx^{4}} + \frac{6{\frac{1}{x}}^{e}{a}^{x}e}{dx^{4}} - \frac{6b{\frac{1}{x}}^{e}e}{dx^{4}} - \frac{11b{\frac{1}{x}}^{e}e^{2}}{dx^{4}} - \frac{6b{\frac{1}{x}}^{e}e^{3}}{dx^{4}} - \frac{b{\frac{1}{x}}^{e}e^{4}}{dx^{4}} - \frac{2c{\frac{1}{x}}^{e}e^{3}}{dx^{3}} - \frac{c{\frac{1}{x}}^{e}e^{4}}{dx^{3}} + \frac{2c{\frac{1}{x}}^{e}e}{dx^{3}} + \frac{c{\frac{1}{x}}^{e}e^{2}}{dx^{3}}\\ \end{split}\end{equation} \]
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