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

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