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。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{({a}^{x} - b - cx){x}^{e}}{d} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{{a}^{x}{x}^{e}}{d} - \frac{b{x}^{e}}{d} - \frac{cx{x}^{e}}{d}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{{a}^{x}{x}^{e}}{d} - \frac{b{x}^{e}}{d} - \frac{cx{x}^{e}}{d}\right)}{dx}\\=&\frac{({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){x}^{e}}{d} + \frac{{a}^{x}({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))}{d} - \frac{b({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))}{d} - \frac{c{x}^{e}}{d} - \frac{cx({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))}{d}\\=&\frac{{a}^{x}{x}^{e}ln(a)}{d} + \frac{{x}^{e}{a}^{x}e}{dx} - \frac{b{x}^{e}e}{dx} - \frac{c{x}^{e}e}{d} - \frac{c{x}^{e}}{d}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{{a}^{x}{x}^{e}ln(a)}{d} + \frac{{x}^{e}{a}^{x}e}{dx} - \frac{b{x}^{e}e}{dx} - \frac{c{x}^{e}e}{d} - \frac{c{x}^{e}}{d}\right)}{dx}\\=&\frac{({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){x}^{e}ln(a)}{d} + \frac{{a}^{x}({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))ln(a)}{d} + \frac{{a}^{x}{x}^{e}*0}{d(a)} + \frac{-{x}^{e}{a}^{x}e}{dx^{2}} + \frac{({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)})){a}^{x}e}{dx} + \frac{{x}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e}{dx} + \frac{{x}^{e}{a}^{x}*0}{dx} - \frac{b*-{x}^{e}e}{dx^{2}} - \frac{b({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))e}{dx} - \frac{b{x}^{e}*0}{dx} - \frac{c({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))e}{d} - \frac{c{x}^{e}*0}{d} - \frac{c({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))}{d}\\=&\frac{{a}^{x}{x}^{e}ln^{2}(a)}{d} + \frac{{x}^{e}{a}^{x}eln(a)}{dx} + \frac{{a}^{x}{x}^{e}eln(a)}{dx} + \frac{{x}^{e}{a}^{x}e^{2}}{dx^{2}} - \frac{{x}^{e}{a}^{x}e}{dx^{2}} + \frac{b{x}^{e}e}{dx^{2}} - \frac{b{x}^{e}e^{2}}{dx^{2}} - \frac{c{x}^{e}e^{2}}{dx} - \frac{c{x}^{e}e}{dx}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{{a}^{x}{x}^{e}ln^{2}(a)}{d} + \frac{{x}^{e}{a}^{x}eln(a)}{dx} + \frac{{a}^{x}{x}^{e}eln(a)}{dx} + \frac{{x}^{e}{a}^{x}e^{2}}{dx^{2}} - \frac{{x}^{e}{a}^{x}e}{dx^{2}} + \frac{b{x}^{e}e}{dx^{2}} - \frac{b{x}^{e}e^{2}}{dx^{2}} - \frac{c{x}^{e}e^{2}}{dx} - \frac{c{x}^{e}e}{dx}\right)}{dx}\\=&\frac{({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){x}^{e}ln^{2}(a)}{d} + \frac{{a}^{x}({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))ln^{2}(a)}{d} + \frac{{a}^{x}{x}^{e}*2ln(a)*0}{d(a)} + \frac{-{x}^{e}{a}^{x}eln(a)}{dx^{2}} + \frac{({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)})){a}^{x}eln(a)}{dx} + \frac{{x}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))eln(a)}{dx} + \frac{{x}^{e}{a}^{x}*0ln(a)}{dx} + \frac{{x}^{e}{a}^{x}e*0}{dx(a)} + \frac{-{a}^{x}{x}^{e}eln(a)}{dx^{2}} + \frac{({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){x}^{e}eln(a)}{dx} + \frac{{a}^{x}({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))eln(a)}{dx} + \frac{{a}^{x}{x}^{e}*0ln(a)}{dx} + \frac{{a}^{x}{x}^{e}e*0}{dx(a)} + \frac{-2{x}^{e}{a}^{x}e^{2}}{dx^{3}} + \frac{({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)})){a}^{x}e^{2}}{dx^{2}} + \frac{{x}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e^{2}}{dx^{2}} + \frac{{x}^{e}{a}^{x}*2e*0}{dx^{2}} - \frac{-2{x}^{e}{a}^{x}e}{dx^{3}} - \frac{({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)})){a}^{x}e}{dx^{2}} - \frac{{x}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e}{dx^{2}} - \frac{{x}^{e}{a}^{x}*0}{dx^{2}} + \frac{b*-2{x}^{e}e}{dx^{3}} + \frac{b({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))e}{dx^{2}} + \frac{b{x}^{e}*0}{dx^{2}} - \frac{b*-2{x}^{e}e^{2}}{dx^{3}} - \frac{b({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))e^{2}}{dx^{2}} - \frac{b{x}^{e}*2e*0}{dx^{2}} - \frac{c*-{x}^{e}e^{2}}{dx^{2}} - \frac{c({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))e^{2}}{dx} - \frac{c{x}^{e}*2e*0}{dx} - \frac{c*-{x}^{e}e}{dx^{2}} - \frac{c({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))e}{dx} - \frac{c{x}^{e}*0}{dx}\\=&\frac{{a}^{x}{x}^{e}ln^{3}(a)}{d} - \frac{{x}^{e}{a}^{x}eln(a)}{dx^{2}} + \frac{2{a}^{x}{x}^{e}eln^{2}(a)}{dx} + \frac{2{x}^{e}{a}^{x}e^{2}ln(a)}{dx^{2}} + \frac{{x}^{e}{a}^{x}eln^{2}(a)}{dx} - \frac{2{a}^{x}{x}^{e}eln(a)}{dx^{2}} + \frac{{a}^{x}{x}^{e}e^{2}ln(a)}{dx^{2}} + \frac{{x}^{e}{a}^{x}e^{3}}{dx^{3}} - \frac{3{x}^{e}{a}^{x}e^{2}}{dx^{3}} + \frac{2{x}^{e}{a}^{x}e}{dx^{3}} - \frac{2b{x}^{e}e}{dx^{3}} + \frac{3b{x}^{e}e^{2}}{dx^{3}} - \frac{b{x}^{e}e^{3}}{dx^{3}} - \frac{c{x}^{e}e^{3}}{dx^{2}} + \frac{c{x}^{e}e}{dx^{2}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{{a}^{x}{x}^{e}ln^{3}(a)}{d} - \frac{{x}^{e}{a}^{x}eln(a)}{dx^{2}} + \frac{2{a}^{x}{x}^{e}eln^{2}(a)}{dx} + \frac{2{x}^{e}{a}^{x}e^{2}ln(a)}{dx^{2}} + \frac{{x}^{e}{a}^{x}eln^{2}(a)}{dx} - \frac{2{a}^{x}{x}^{e}eln(a)}{dx^{2}} + \frac{{a}^{x}{x}^{e}e^{2}ln(a)}{dx^{2}} + \frac{{x}^{e}{a}^{x}e^{3}}{dx^{3}} - \frac{3{x}^{e}{a}^{x}e^{2}}{dx^{3}} + \frac{2{x}^{e}{a}^{x}e}{dx^{3}} - \frac{2b{x}^{e}e}{dx^{3}} + \frac{3b{x}^{e}e^{2}}{dx^{3}} - \frac{b{x}^{e}e^{3}}{dx^{3}} - \frac{c{x}^{e}e^{3}}{dx^{2}} + \frac{c{x}^{e}e}{dx^{2}}\right)}{dx}\\=&\frac{({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){x}^{e}ln^{3}(a)}{d} + \frac{{a}^{x}({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))ln^{3}(a)}{d} + \frac{{a}^{x}{x}^{e}*3ln^{2}(a)*0}{d(a)} - \frac{-2{x}^{e}{a}^{x}eln(a)}{dx^{3}} - \frac{({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)})){a}^{x}eln(a)}{dx^{2}} - \frac{{x}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))eln(a)}{dx^{2}} - \frac{{x}^{e}{a}^{x}*0ln(a)}{dx^{2}} - \frac{{x}^{e}{a}^{x}e*0}{dx^{2}(a)} + \frac{2*-{a}^{x}{x}^{e}eln^{2}(a)}{dx^{2}} + \frac{2({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){x}^{e}eln^{2}(a)}{dx} + \frac{2{a}^{x}({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))eln^{2}(a)}{dx} + \frac{2{a}^{x}{x}^{e}*0ln^{2}(a)}{dx} + \frac{2{a}^{x}{x}^{e}e*2ln(a)*0}{dx(a)} + \frac{2*-2{x}^{e}{a}^{x}e^{2}ln(a)}{dx^{3}} + \frac{2({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)})){a}^{x}e^{2}ln(a)}{dx^{2}} + \frac{2{x}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e^{2}ln(a)}{dx^{2}} + \frac{2{x}^{e}{a}^{x}*2e*0ln(a)}{dx^{2}} + \frac{2{x}^{e}{a}^{x}e^{2}*0}{dx^{2}(a)} + \frac{-{x}^{e}{a}^{x}eln^{2}(a)}{dx^{2}} + \frac{({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)})){a}^{x}eln^{2}(a)}{dx} + \frac{{x}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))eln^{2}(a)}{dx} + \frac{{x}^{e}{a}^{x}*0ln^{2}(a)}{dx} + \frac{{x}^{e}{a}^{x}e*2ln(a)*0}{dx(a)} - \frac{2*-2{a}^{x}{x}^{e}eln(a)}{dx^{3}} - \frac{2({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){x}^{e}eln(a)}{dx^{2}} - \frac{2{a}^{x}({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))eln(a)}{dx^{2}} - \frac{2{a}^{x}{x}^{e}*0ln(a)}{dx^{2}} - \frac{2{a}^{x}{x}^{e}e*0}{dx^{2}(a)} + \frac{-2{a}^{x}{x}^{e}e^{2}ln(a)}{dx^{3}} + \frac{({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)})){x}^{e}e^{2}ln(a)}{dx^{2}} + \frac{{a}^{x}({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))e^{2}ln(a)}{dx^{2}} + \frac{{a}^{x}{x}^{e}*2e*0ln(a)}{dx^{2}} + \frac{{a}^{x}{x}^{e}e^{2}*0}{dx^{2}(a)} + \frac{-3{x}^{e}{a}^{x}e^{3}}{dx^{4}} + \frac{({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)})){a}^{x}e^{3}}{dx^{3}} + \frac{{x}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e^{3}}{dx^{3}} + \frac{{x}^{e}{a}^{x}*3e^{2}*0}{dx^{3}} - \frac{3*-3{x}^{e}{a}^{x}e^{2}}{dx^{4}} - \frac{3({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)})){a}^{x}e^{2}}{dx^{3}} - \frac{3{x}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e^{2}}{dx^{3}} - \frac{3{x}^{e}{a}^{x}*2e*0}{dx^{3}} + \frac{2*-3{x}^{e}{a}^{x}e}{dx^{4}} + \frac{2({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)})){a}^{x}e}{dx^{3}} + \frac{2{x}^{e}({a}^{x}((1)ln(a) + \frac{(x)(0)}{(a)}))e}{dx^{3}} + \frac{2{x}^{e}{a}^{x}*0}{dx^{3}} - \frac{2b*-3{x}^{e}e}{dx^{4}} - \frac{2b({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))e}{dx^{3}} - \frac{2b{x}^{e}*0}{dx^{3}} + \frac{3b*-3{x}^{e}e^{2}}{dx^{4}} + \frac{3b({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))e^{2}}{dx^{3}} + \frac{3b{x}^{e}*2e*0}{dx^{3}} - \frac{b*-3{x}^{e}e^{3}}{dx^{4}} - \frac{b({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))e^{3}}{dx^{3}} - \frac{b{x}^{e}*3e^{2}*0}{dx^{3}} - \frac{c*-2{x}^{e}e^{3}}{dx^{3}} - \frac{c({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))e^{3}}{dx^{2}} - \frac{c{x}^{e}*3e^{2}*0}{dx^{2}} + \frac{c*-2{x}^{e}e}{dx^{3}} + \frac{c({x}^{e}((0)ln(x) + \frac{(e)(1)}{(x)}))e}{dx^{2}} + \frac{c{x}^{e}*0}{dx^{2}}\\=&\frac{{a}^{x}{x}^{e}ln^{4}(a)}{d} + \frac{2{x}^{e}{a}^{x}eln(a)}{dx^{3}} - \frac{5{a}^{x}{x}^{e}eln^{2}(a)}{dx^{2}} - \frac{7{x}^{e}{a}^{x}e^{2}ln(a)}{dx^{3}} + \frac{3{a}^{x}{x}^{e}eln^{3}(a)}{dx} + \frac{3{x}^{e}{a}^{x}e^{2}ln^{2}(a)}{dx^{2}} + \frac{3{a}^{x}{x}^{e}e^{2}ln^{2}(a)}{dx^{2}} + \frac{3{x}^{e}{a}^{x}e^{3}ln(a)}{dx^{3}} + \frac{{x}^{e}{a}^{x}eln^{3}(a)}{dx} - \frac{{x}^{e}{a}^{x}eln^{2}(a)}{dx^{2}} + \frac{6{a}^{x}{x}^{e}eln(a)}{dx^{3}} - \frac{5{a}^{x}{x}^{e}e^{2}ln(a)}{dx^{3}} + \frac{{a}^{x}{x}^{e}e^{3}ln(a)}{dx^{3}} + \frac{{x}^{e}{a}^{x}e^{4}}{dx^{4}} - \frac{6{x}^{e}{a}^{x}e^{3}}{dx^{4}} + \frac{11{x}^{e}{a}^{x}e^{2}}{dx^{4}} - \frac{6{x}^{e}{a}^{x}e}{dx^{4}} + \frac{6b{x}^{e}e}{dx^{4}} - \frac{11b{x}^{e}e^{2}}{dx^{4}} + \frac{6b{x}^{e}e^{3}}{dx^{4}} - \frac{b{x}^{e}e^{4}}{dx^{4}} + \frac{2c{x}^{e}e^{3}}{dx^{3}} - \frac{c{x}^{e}e^{4}}{dx^{3}} - \frac{2c{x}^{e}e}{dx^{3}} + \frac{c{x}^{e}e^{2}}{dx^{3}}\\ \end{split}\end{equation} \]

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