There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {e}^{x}ln(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {e}^{x}ln(x)\right)}{dx}\\=&({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(x) + \frac{{e}^{x}}{(x)}\\=&{e}^{x}ln(x) + \frac{{e}^{x}}{x}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( {e}^{x}ln(x) + \frac{{e}^{x}}{x}\right)}{dx}\\=&({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(x) + \frac{{e}^{x}}{(x)} + \frac{-{e}^{x}}{x^{2}} + \frac{({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{x}\\=&{e}^{x}ln(x) + \frac{2{e}^{x}}{x} - \frac{{e}^{x}}{x^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( {e}^{x}ln(x) + \frac{2{e}^{x}}{x} - \frac{{e}^{x}}{x^{2}}\right)}{dx}\\=&({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(x) + \frac{{e}^{x}}{(x)} + \frac{2*-{e}^{x}}{x^{2}} + \frac{2({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{x} - \frac{-2{e}^{x}}{x^{3}} - \frac{({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{x^{2}}\\=&{e}^{x}ln(x) + \frac{3{e}^{x}}{x} - \frac{3{e}^{x}}{x^{2}} + \frac{2{e}^{x}}{x^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( {e}^{x}ln(x) + \frac{3{e}^{x}}{x} - \frac{3{e}^{x}}{x^{2}} + \frac{2{e}^{x}}{x^{3}}\right)}{dx}\\=&({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(x) + \frac{{e}^{x}}{(x)} + \frac{3*-{e}^{x}}{x^{2}} + \frac{3({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{x} - \frac{3*-2{e}^{x}}{x^{3}} - \frac{3({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{x^{2}} + \frac{2*-3{e}^{x}}{x^{4}} + \frac{2({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{x^{3}}\\=&{e}^{x}ln(x) + \frac{4{e}^{x}}{x} - \frac{6{e}^{x}}{x^{2}} + \frac{8{e}^{x}}{x^{3}} - \frac{6{e}^{x}}{x^{4}}\\ \end{split}\end{equation} \]

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