There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{x}{sqrt(x + 1)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x}{sqrt(x + 1)}\right)}{dx}\\=&\frac{1}{sqrt(x + 1)} + \frac{x*-(1 + 0)*\frac{1}{2}}{(x + 1)(x + 1)^{\frac{1}{2}}}\\=&\frac{1}{sqrt(x + 1)} - \frac{x}{2(x + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{sqrt(x + 1)} - \frac{x}{2(x + 1)^{\frac{3}{2}}}\right)}{dx}\\=&\frac{-(1 + 0)*\frac{1}{2}}{(x + 1)(x + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-3}{2}(1 + 0)}{(x + 1)^{\frac{5}{2}}})x}{2} - \frac{1}{2(x + 1)^{\frac{3}{2}}}\\=&\frac{3x}{4(x + 1)^{\frac{5}{2}}} - \frac{1}{(x + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{3x}{4(x + 1)^{\frac{5}{2}}} - \frac{1}{(x + 1)^{\frac{3}{2}}}\right)}{dx}\\=&\frac{3(\frac{\frac{-5}{2}(1 + 0)}{(x + 1)^{\frac{7}{2}}})x}{4} + \frac{3}{4(x + 1)^{\frac{5}{2}}} - (\frac{\frac{-3}{2}(1 + 0)}{(x + 1)^{\frac{5}{2}}})\\=& - \frac{15x}{8(x + 1)^{\frac{7}{2}}} + \frac{9}{4(x + 1)^{\frac{5}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - \frac{15x}{8(x + 1)^{\frac{7}{2}}} + \frac{9}{4(x + 1)^{\frac{5}{2}}}\right)}{dx}\\=& - \frac{15(\frac{\frac{-7}{2}(1 + 0)}{(x + 1)^{\frac{9}{2}}})x}{8} - \frac{15}{8(x + 1)^{\frac{7}{2}}} + \frac{9(\frac{\frac{-5}{2}(1 + 0)}{(x + 1)^{\frac{7}{2}}})}{4}\\=&\frac{105x}{16(x + 1)^{\frac{9}{2}}} - \frac{15}{2(x + 1)^{\frac{7}{2}}}\\ \end{split}\end{equation} \]

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