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 2 个题目：每一题对 x 求 4 阶导数。
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/2】求函数{x}^{X} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( {x}^{X}\right)}{dx}\\=&({x}^{X}((0)ln(x) + \frac{(X)(1)}{(x)}))\\=&\frac{X{x}^{X}}{x}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{X{x}^{X}}{x}\right)}{dx}\\=&\frac{X*-{x}^{X}}{x^{2}} + \frac{X({x}^{X}((0)ln(x) + \frac{(X)(1)}{(x)}))}{x}\\=&\frac{-X{x}^{X}}{x^{2}} + \frac{X^{2}{x}^{X}}{x^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-X{x}^{X}}{x^{2}} + \frac{X^{2}{x}^{X}}{x^{2}}\right)}{dx}\\=&\frac{-X*-2{x}^{X}}{x^{3}} - \frac{X({x}^{X}((0)ln(x) + \frac{(X)(1)}{(x)}))}{x^{2}} + \frac{X^{2}*-2{x}^{X}}{x^{3}} + \frac{X^{2}({x}^{X}((0)ln(x) + \frac{(X)(1)}{(x)}))}{x^{2}}\\=&\frac{2X{x}^{X}}{x^{3}} - \frac{3X^{2}{x}^{X}}{x^{3}} + \frac{X^{3}{x}^{X}}{x^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{2X{x}^{X}}{x^{3}} - \frac{3X^{2}{x}^{X}}{x^{3}} + \frac{X^{3}{x}^{X}}{x^{3}}\right)}{dx}\\=&\frac{2X*-3{x}^{X}}{x^{4}} + \frac{2X({x}^{X}((0)ln(x) + \frac{(X)(1)}{(x)}))}{x^{3}} - \frac{3X^{2}*-3{x}^{X}}{x^{4}} - \frac{3X^{2}({x}^{X}((0)ln(x) + \frac{(X)(1)}{(x)}))}{x^{3}} + \frac{X^{3}*-3{x}^{X}}{x^{4}} + \frac{X^{3}({x}^{X}((0)ln(x) + \frac{(X)(1)}{(x)}))}{x^{3}}\\=&\frac{-6X{x}^{X}}{x^{4}} + \frac{11X^{2}{x}^{X}}{x^{4}} - \frac{6X^{3}{x}^{X}}{x^{4}} + \frac{X^{4}{x}^{X}}{x^{4}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【2/2】求函数{X}^{x}{(ln(x))}^{4} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = {X}^{x}ln^{4}(x)\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( {X}^{x}ln^{4}(x)\right)}{dx}\\=&({X}^{x}((1)ln(X) + \frac{(x)(0)}{(X)}))ln^{4}(x) + \frac{{X}^{x}*4ln^{3}(x)}{(x)}\\=&{X}^{x}ln(X)ln^{4}(x) + \frac{4{X}^{x}ln^{3}(x)}{x}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( {X}^{x}ln(X)ln^{4}(x) + \frac{4{X}^{x}ln^{3}(x)}{x}\right)}{dx}\\=&({X}^{x}((1)ln(X) + \frac{(x)(0)}{(X)}))ln(X)ln^{4}(x) + \frac{{X}^{x}*0ln^{4}(x)}{(X)} + \frac{{X}^{x}ln(X)*4ln^{3}(x)}{(x)} + \frac{4*-{X}^{x}ln^{3}(x)}{x^{2}} + \frac{4({X}^{x}((1)ln(X) + \frac{(x)(0)}{(X)}))ln^{3}(x)}{x} + \frac{4{X}^{x}*3ln^{2}(x)}{x(x)}\\=&{X}^{x}ln^{2}(X)ln^{4}(x) + \frac{4{X}^{x}ln^{3}(x)ln(X)}{x} + \frac{4{X}^{x}ln(X)ln^{3}(x)}{x} - \frac{4{X}^{x}ln^{3}(x)}{x^{2}} + \frac{12{X}^{x}ln^{2}(x)}{x^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( {X}^{x}ln^{2}(X)ln^{4}(x) + \frac{4{X}^{x}ln^{3}(x)ln(X)}{x} + \frac{4{X}^{x}ln(X)ln^{3}(x)}{x} - \frac{4{X}^{x}ln^{3}(x)}{x^{2}} + \frac{12{X}^{x}ln^{2}(x)}{x^{2}}\right)}{dx}\\=&({X}^{x}((1)ln(X) + \frac{(x)(0)}{(X)}))ln^{2}(X)ln^{4}(x) + \frac{{X}^{x}*2ln(X)*0ln^{4}(x)}{(X)} + \frac{{X}^{x}ln^{2}(X)*4ln^{3}(x)}{(x)} + \frac{4*-{X}^{x}ln^{3}(x)ln(X)}{x^{2}} + \frac{4({X}^{x}((1)ln(X) + \frac{(x)(0)}{(X)}))ln^{3}(x)ln(X)}{x} + \frac{4{X}^{x}*3ln^{2}(x)ln(X)}{x(x)} + \frac{4{X}^{x}ln^{3}(x)*0}{x(X)} + \frac{4*-{X}^{x}ln(X)ln^{3}(x)}{x^{2}} + \frac{4({X}^{x}((1)ln(X) + \frac{(x)(0)}{(X)}))ln(X)ln^{3}(x)}{x} + \frac{4{X}^{x}*0ln^{3}(x)}{x(X)} + \frac{4{X}^{x}ln(X)*3ln^{2}(x)}{x(x)} - \frac{4*-2{X}^{x}ln^{3}(x)}{x^{3}} - \frac{4({X}^{x}((1)ln(X) + \frac{(x)(0)}{(X)}))ln^{3}(x)}{x^{2}} - \frac{4{X}^{x}*3ln^{2}(x)}{x^{2}(x)} + \frac{12*-2{X}^{x}ln^{2}(x)}{x^{3}} + \frac{12({X}^{x}((1)ln(X) + \frac{(x)(0)}{(X)}))ln^{2}(x)}{x^{2}} + \frac{12{X}^{x}*2ln(x)}{x^{2}(x)}\\=&{X}^{x}ln^{3}(X)ln^{4}(x) + \frac{4{X}^{x}ln^{3}(x)ln^{2}(X)}{x} - \frac{4{X}^{x}ln^{3}(x)ln(X)}{x^{2}} + \frac{8{X}^{x}ln^{2}(X)ln^{3}(x)}{x} + \frac{24{X}^{x}ln^{2}(x)ln(X)}{x^{2}} - \frac{8{X}^{x}ln(X)ln^{3}(x)}{x^{2}} + \frac{12{X}^{x}ln(X)ln^{2}(x)}{x^{2}} - \frac{36{X}^{x}ln^{2}(x)}{x^{3}} + \frac{8{X}^{x}ln^{3}(x)}{x^{3}} + \frac{24{X}^{x}ln(x)}{x^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( {X}^{x}ln^{3}(X)ln^{4}(x) + \frac{4{X}^{x}ln^{3}(x)ln^{2}(X)}{x} - \frac{4{X}^{x}ln^{3}(x)ln(X)}{x^{2}} + \frac{8{X}^{x}ln^{2}(X)ln^{3}(x)}{x} + \frac{24{X}^{x}ln^{2}(x)ln(X)}{x^{2}} - \frac{8{X}^{x}ln(X)ln^{3}(x)}{x^{2}} + \frac{12{X}^{x}ln(X)ln^{2}(x)}{x^{2}} - \frac{36{X}^{x}ln^{2}(x)}{x^{3}} + \frac{8{X}^{x}ln^{3}(x)}{x^{3}} + \frac{24{X}^{x}ln(x)}{x^{3}}\right)}{dx}\\=&({X}^{x}((1)ln(X) + \frac{(x)(0)}{(X)}))ln^{3}(X)ln^{4}(x) + \frac{{X}^{x}*3ln^{2}(X)*0ln^{4}(x)}{(X)} + \frac{{X}^{x}ln^{3}(X)*4ln^{3}(x)}{(x)} + \frac{4*-{X}^{x}ln^{3}(x)ln^{2}(X)}{x^{2}} + \frac{4({X}^{x}((1)ln(X) + \frac{(x)(0)}{(X)}))ln^{3}(x)ln^{2}(X)}{x} + \frac{4{X}^{x}*3ln^{2}(x)ln^{2}(X)}{x(x)} + \frac{4{X}^{x}ln^{3}(x)*2ln(X)*0}{x(X)} - \frac{4*-2{X}^{x}ln^{3}(x)ln(X)}{x^{3}} - \frac{4({X}^{x}((1)ln(X) + \frac{(x)(0)}{(X)}))ln^{3}(x)ln(X)}{x^{2}} - \frac{4{X}^{x}*3ln^{2}(x)ln(X)}{x^{2}(x)} - \frac{4{X}^{x}ln^{3}(x)*0}{x^{2}(X)} + \frac{8*-{X}^{x}ln^{2}(X)ln^{3}(x)}{x^{2}} + \frac{8({X}^{x}((1)ln(X) + \frac{(x)(0)}{(X)}))ln^{2}(X)ln^{3}(x)}{x} + \frac{8{X}^{x}*2ln(X)*0ln^{3}(x)}{x(X)} + \frac{8{X}^{x}ln^{2}(X)*3ln^{2}(x)}{x(x)} + \frac{24*-2{X}^{x}ln^{2}(x)ln(X)}{x^{3}} + \frac{24({X}^{x}((1)ln(X) + \frac{(x)(0)}{(X)}))ln^{2}(x)ln(X)}{x^{2}} + \frac{24{X}^{x}*2ln(x)ln(X)}{x^{2}(x)} + \frac{24{X}^{x}ln^{2}(x)*0}{x^{2}(X)} - \frac{8*-2{X}^{x}ln(X)ln^{3}(x)}{x^{3}} - \frac{8({X}^{x}((1)ln(X) + \frac{(x)(0)}{(X)}))ln(X)ln^{3}(x)}{x^{2}} - \frac{8{X}^{x}*0ln^{3}(x)}{x^{2}(X)} - \frac{8{X}^{x}ln(X)*3ln^{2}(x)}{x^{2}(x)} + \frac{12*-2{X}^{x}ln(X)ln^{2}(x)}{x^{3}} + \frac{12({X}^{x}((1)ln(X) + \frac{(x)(0)}{(X)}))ln(X)ln^{2}(x)}{x^{2}} + \frac{12{X}^{x}*0ln^{2}(x)}{x^{2}(X)} + \frac{12{X}^{x}ln(X)*2ln(x)}{x^{2}(x)} - \frac{36*-3{X}^{x}ln^{2}(x)}{x^{4}} - \frac{36({X}^{x}((1)ln(X) + \frac{(x)(0)}{(X)}))ln^{2}(x)}{x^{3}} - \frac{36{X}^{x}*2ln(x)}{x^{3}(x)} + \frac{8*-3{X}^{x}ln^{3}(x)}{x^{4}} + \frac{8({X}^{x}((1)ln(X) + \frac{(x)(0)}{(X)}))ln^{3}(x)}{x^{3}} + \frac{8{X}^{x}*3ln^{2}(x)}{x^{3}(x)} + \frac{24*-3{X}^{x}ln(x)}{x^{4}} + \frac{24({X}^{x}((1)ln(X) + \frac{(x)(0)}{(X)}))ln(x)}{x^{3}} + \frac{24{X}^{x}}{x^{3}(x)}\\=&{X}^{x}ln^{4}(X)ln^{4}(x) + \frac{4{X}^{x}ln^{3}(x)ln^{3}(X)}{x} - \frac{4{X}^{x}ln^{3}(x)ln^{2}(X)}{x^{2}} + \frac{12{X}^{x}ln^{3}(X)ln^{3}(x)}{x} + \frac{36{X}^{x}ln^{2}(x)ln^{2}(X)}{x^{2}} + \frac{8{X}^{x}ln^{3}(x)ln(X)}{x^{3}} - \frac{20{X}^{x}ln^{2}(X)ln^{3}(x)}{x^{2}} - \frac{84{X}^{x}ln^{2}(x)ln(X)}{x^{3}} + \frac{36{X}^{x}ln^{2}(X)ln^{2}(x)}{x^{2}} + \frac{72{X}^{x}ln(x)ln(X)}{x^{3}} + \frac{24{X}^{x}ln(X)ln^{3}(x)}{x^{3}} - \frac{60{X}^{x}ln(X)ln^{2}(x)}{x^{3}} + \frac{24{X}^{x}ln(X)ln(x)}{x^{3}} - \frac{144{X}^{x}ln(x)}{x^{4}} - \frac{24{X}^{x}ln^{3}(x)}{x^{4}} + \frac{132{X}^{x}ln^{2}(x)}{x^{4}} + \frac{24{X}^{x}}{x^{4}}\\ \end{split}\end{equation} \]
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