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

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