There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
Note that variables are case sensitive.\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{X}{(1 - {e}^{\frac{1}{x}})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]
\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{X}{(-{e}^{\frac{1}{x}} + 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{X}{(-{e}^{\frac{1}{x}} + 1)}\right)}{dx}\\=&(\frac{-(-({e}^{\frac{1}{x}}((\frac{-1}{x^{2}})ln(e) + \frac{(\frac{1}{x})(0)}{(e)})) + 0)}{(-{e}^{\frac{1}{x}} + 1)^{2}})X + 0\\=&\frac{-X{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-X{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{2}}\right)}{dx}\\=&\frac{-(\frac{-2(-({e}^{\frac{1}{x}}((\frac{-1}{x^{2}})ln(e) + \frac{(\frac{1}{x})(0)}{(e)})) + 0)}{(-{e}^{\frac{1}{x}} + 1)^{3}})X{e}^{\frac{1}{x}}}{x^{2}} - \frac{X*-2{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{3}} - \frac{X({e}^{\frac{1}{x}}((\frac{-1}{x^{2}})ln(e) + \frac{(\frac{1}{x})(0)}{(e)}))}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{2}}\\=&\frac{2X{e}^{(\frac{2}{x})}}{(-{e}^{\frac{1}{x}} + 1)^{3}x^{4}} + \frac{2X{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{3}} + \frac{X{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{4}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2X{e}^{(\frac{2}{x})}}{(-{e}^{\frac{1}{x}} + 1)^{3}x^{4}} + \frac{2X{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{3}} + \frac{X{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{4}}\right)}{dx}\\=&\frac{2(\frac{-3(-({e}^{\frac{1}{x}}((\frac{-1}{x^{2}})ln(e) + \frac{(\frac{1}{x})(0)}{(e)})) + 0)}{(-{e}^{\frac{1}{x}} + 1)^{4}})X{e}^{(\frac{2}{x})}}{x^{4}} + \frac{2X*-4{e}^{(\frac{2}{x})}}{(-{e}^{\frac{1}{x}} + 1)^{3}x^{5}} + \frac{2X({e}^{(\frac{2}{x})}((\frac{2*-1}{x^{2}})ln(e) + \frac{(\frac{2}{x})(0)}{(e)}))}{(-{e}^{\frac{1}{x}} + 1)^{3}x^{4}} + \frac{2(\frac{-2(-({e}^{\frac{1}{x}}((\frac{-1}{x^{2}})ln(e) + \frac{(\frac{1}{x})(0)}{(e)})) + 0)}{(-{e}^{\frac{1}{x}} + 1)^{3}})X{e}^{\frac{1}{x}}}{x^{3}} + \frac{2X*-3{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{4}} + \frac{2X({e}^{\frac{1}{x}}((\frac{-1}{x^{2}})ln(e) + \frac{(\frac{1}{x})(0)}{(e)}))}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{3}} + \frac{(\frac{-2(-({e}^{\frac{1}{x}}((\frac{-1}{x^{2}})ln(e) + \frac{(\frac{1}{x})(0)}{(e)})) + 0)}{(-{e}^{\frac{1}{x}} + 1)^{3}})X{e}^{\frac{1}{x}}}{x^{4}} + \frac{X*-4{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{5}} + \frac{X({e}^{\frac{1}{x}}((\frac{-1}{x^{2}})ln(e) + \frac{(\frac{1}{x})(0)}{(e)}))}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{4}}\\=&\frac{-6X{e}^{(\frac{3}{x})}}{(-{e}^{\frac{1}{x}} + 1)^{4}x^{6}} - \frac{12X{e}^{(\frac{2}{x})}}{(-{e}^{\frac{1}{x}} + 1)^{3}x^{5}} - \frac{6X{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{4}} - \frac{6X{e}^{(\frac{2}{x})}}{(-{e}^{\frac{1}{x}} + 1)^{3}x^{6}} - \frac{6X{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{5}} - \frac{X{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{6}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6X{e}^{(\frac{3}{x})}}{(-{e}^{\frac{1}{x}} + 1)^{4}x^{6}} - \frac{12X{e}^{(\frac{2}{x})}}{(-{e}^{\frac{1}{x}} + 1)^{3}x^{5}} - \frac{6X{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{4}} - \frac{6X{e}^{(\frac{2}{x})}}{(-{e}^{\frac{1}{x}} + 1)^{3}x^{6}} - \frac{6X{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{5}} - \frac{X{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{6}}\right)}{dx}\\=&\frac{-6(\frac{-4(-({e}^{\frac{1}{x}}((\frac{-1}{x^{2}})ln(e) + \frac{(\frac{1}{x})(0)}{(e)})) + 0)}{(-{e}^{\frac{1}{x}} + 1)^{5}})X{e}^{(\frac{3}{x})}}{x^{6}} - \frac{6X*-6{e}^{(\frac{3}{x})}}{(-{e}^{\frac{1}{x}} + 1)^{4}x^{7}} - \frac{6X({e}^{(\frac{3}{x})}((\frac{3*-1}{x^{2}})ln(e) + \frac{(\frac{3}{x})(0)}{(e)}))}{(-{e}^{\frac{1}{x}} + 1)^{4}x^{6}} - \frac{12(\frac{-3(-({e}^{\frac{1}{x}}((\frac{-1}{x^{2}})ln(e) + \frac{(\frac{1}{x})(0)}{(e)})) + 0)}{(-{e}^{\frac{1}{x}} + 1)^{4}})X{e}^{(\frac{2}{x})}}{x^{5}} - \frac{12X*-5{e}^{(\frac{2}{x})}}{(-{e}^{\frac{1}{x}} + 1)^{3}x^{6}} - \frac{12X({e}^{(\frac{2}{x})}((\frac{2*-1}{x^{2}})ln(e) + \frac{(\frac{2}{x})(0)}{(e)}))}{(-{e}^{\frac{1}{x}} + 1)^{3}x^{5}} - \frac{6(\frac{-2(-({e}^{\frac{1}{x}}((\frac{-1}{x^{2}})ln(e) + \frac{(\frac{1}{x})(0)}{(e)})) + 0)}{(-{e}^{\frac{1}{x}} + 1)^{3}})X{e}^{\frac{1}{x}}}{x^{4}} - \frac{6X*-4{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{5}} - \frac{6X({e}^{\frac{1}{x}}((\frac{-1}{x^{2}})ln(e) + \frac{(\frac{1}{x})(0)}{(e)}))}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{4}} - \frac{6(\frac{-3(-({e}^{\frac{1}{x}}((\frac{-1}{x^{2}})ln(e) + \frac{(\frac{1}{x})(0)}{(e)})) + 0)}{(-{e}^{\frac{1}{x}} + 1)^{4}})X{e}^{(\frac{2}{x})}}{x^{6}} - \frac{6X*-6{e}^{(\frac{2}{x})}}{(-{e}^{\frac{1}{x}} + 1)^{3}x^{7}} - \frac{6X({e}^{(\frac{2}{x})}((\frac{2*-1}{x^{2}})ln(e) + \frac{(\frac{2}{x})(0)}{(e)}))}{(-{e}^{\frac{1}{x}} + 1)^{3}x^{6}} - \frac{6(\frac{-2(-({e}^{\frac{1}{x}}((\frac{-1}{x^{2}})ln(e) + \frac{(\frac{1}{x})(0)}{(e)})) + 0)}{(-{e}^{\frac{1}{x}} + 1)^{3}})X{e}^{\frac{1}{x}}}{x^{5}} - \frac{6X*-5{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{6}} - \frac{6X({e}^{\frac{1}{x}}((\frac{-1}{x^{2}})ln(e) + \frac{(\frac{1}{x})(0)}{(e)}))}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{5}} - \frac{(\frac{-2(-({e}^{\frac{1}{x}}((\frac{-1}{x^{2}})ln(e) + \frac{(\frac{1}{x})(0)}{(e)})) + 0)}{(-{e}^{\frac{1}{x}} + 1)^{3}})X{e}^{\frac{1}{x}}}{x^{6}} - \frac{X*-6{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{7}} - \frac{X({e}^{\frac{1}{x}}((\frac{-1}{x^{2}})ln(e) + \frac{(\frac{1}{x})(0)}{(e)}))}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{6}}\\=&\frac{24X{e}^{(\frac{4}{x})}}{(-{e}^{\frac{1}{x}} + 1)^{5}x^{8}} + \frac{72X{e}^{(\frac{3}{x})}}{(-{e}^{\frac{1}{x}} + 1)^{4}x^{7}} + \frac{72X{e}^{(\frac{2}{x})}}{(-{e}^{\frac{1}{x}} + 1)^{3}x^{6}} + \frac{24X{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{5}} + \frac{36X{e}^{(\frac{3}{x})}}{(-{e}^{\frac{1}{x}} + 1)^{4}x^{8}} + \frac{72X{e}^{(\frac{2}{x})}}{(-{e}^{\frac{1}{x}} + 1)^{3}x^{7}} + \frac{14X{e}^{(\frac{2}{x})}}{(-{e}^{\frac{1}{x}} + 1)^{3}x^{8}} + \frac{36X{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{6}} + \frac{12X{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{7}} + \frac{X{e}^{\frac{1}{x}}}{(-{e}^{\frac{1}{x}} + 1)^{2}x^{8}}\\ \end{split}\end{equation} \]Your problem has not been solved here? Please take a look at the hot problems ！
