There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{x}{ln(x + 1)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x}{ln(x + 1)}\right)}{dx}\\=&\frac{1}{ln(x + 1)} + \frac{x*-(1 + 0)}{ln^{2}(x + 1)(x + 1)}\\=&\frac{1}{ln(x + 1)} - \frac{x}{(x + 1)ln^{2}(x + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{ln(x + 1)} - \frac{x}{(x + 1)ln^{2}(x + 1)}\right)}{dx}\\=&\frac{-(1 + 0)}{ln^{2}(x + 1)(x + 1)} - \frac{(\frac{-(1 + 0)}{(x + 1)^{2}})x}{ln^{2}(x + 1)} - \frac{1}{(x + 1)ln^{2}(x + 1)} - \frac{x*-2(1 + 0)}{(x + 1)ln^{3}(x + 1)(x + 1)}\\=&\frac{-2}{(x + 1)ln^{2}(x + 1)} + \frac{x}{(x + 1)^{2}ln^{2}(x + 1)} + \frac{2x}{(x + 1)^{2}ln^{3}(x + 1)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2}{(x + 1)ln^{2}(x + 1)} + \frac{x}{(x + 1)^{2}ln^{2}(x + 1)} + \frac{2x}{(x + 1)^{2}ln^{3}(x + 1)}\right)}{dx}\\=&\frac{-2(\frac{-(1 + 0)}{(x + 1)^{2}})}{ln^{2}(x + 1)} - \frac{2*-2(1 + 0)}{(x + 1)ln^{3}(x + 1)(x + 1)} + \frac{(\frac{-2(1 + 0)}{(x + 1)^{3}})x}{ln^{2}(x + 1)} + \frac{1}{(x + 1)^{2}ln^{2}(x + 1)} + \frac{x*-2(1 + 0)}{(x + 1)^{2}ln^{3}(x + 1)(x + 1)} + \frac{2(\frac{-2(1 + 0)}{(x + 1)^{3}})x}{ln^{3}(x + 1)} + \frac{2}{(x + 1)^{2}ln^{3}(x + 1)} + \frac{2x*-3(1 + 0)}{(x + 1)^{2}ln^{4}(x + 1)(x + 1)}\\=&\frac{3}{(x + 1)^{2}ln^{2}(x + 1)} + \frac{6}{(x + 1)^{2}ln^{3}(x + 1)} - \frac{2x}{(x + 1)^{3}ln^{2}(x + 1)} - \frac{6x}{(x + 1)^{3}ln^{3}(x + 1)} - \frac{6x}{(x + 1)^{3}ln^{4}(x + 1)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3}{(x + 1)^{2}ln^{2}(x + 1)} + \frac{6}{(x + 1)^{2}ln^{3}(x + 1)} - \frac{2x}{(x + 1)^{3}ln^{2}(x + 1)} - \frac{6x}{(x + 1)^{3}ln^{3}(x + 1)} - \frac{6x}{(x + 1)^{3}ln^{4}(x + 1)}\right)}{dx}\\=&\frac{3(\frac{-2(1 + 0)}{(x + 1)^{3}})}{ln^{2}(x + 1)} + \frac{3*-2(1 + 0)}{(x + 1)^{2}ln^{3}(x + 1)(x + 1)} + \frac{6(\frac{-2(1 + 0)}{(x + 1)^{3}})}{ln^{3}(x + 1)} + \frac{6*-3(1 + 0)}{(x + 1)^{2}ln^{4}(x + 1)(x + 1)} - \frac{2(\frac{-3(1 + 0)}{(x + 1)^{4}})x}{ln^{2}(x + 1)} - \frac{2}{(x + 1)^{3}ln^{2}(x + 1)} - \frac{2x*-2(1 + 0)}{(x + 1)^{3}ln^{3}(x + 1)(x + 1)} - \frac{6(\frac{-3(1 + 0)}{(x + 1)^{4}})x}{ln^{3}(x + 1)} - \frac{6}{(x + 1)^{3}ln^{3}(x + 1)} - \frac{6x*-3(1 + 0)}{(x + 1)^{3}ln^{4}(x + 1)(x + 1)} - \frac{6(\frac{-3(1 + 0)}{(x + 1)^{4}})x}{ln^{4}(x + 1)} - \frac{6}{(x + 1)^{3}ln^{4}(x + 1)} - \frac{6x*-4(1 + 0)}{(x + 1)^{3}ln^{5}(x + 1)(x + 1)}\\=&\frac{-8}{(x + 1)^{3}ln^{2}(x + 1)} - \frac{24}{(x + 1)^{3}ln^{3}(x + 1)} - \frac{24}{(x + 1)^{3}ln^{4}(x + 1)} + \frac{6x}{(x + 1)^{4}ln^{2}(x + 1)} + \frac{22x}{(x + 1)^{4}ln^{3}(x + 1)} + \frac{36x}{(x + 1)^{4}ln^{4}(x + 1)} + \frac{24x}{(x + 1)^{4}ln^{5}(x + 1)}\\ \end{split}\end{equation} \]

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