There are 1 questions in this calculation: for each question, the 4 derivative of d is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ ln(d)arccos(d)\ with\ respect\ to\ d：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln(d)arccos(d)\right)}{dd}\\=&\frac{arccos(d)}{(d)} + ln(d)(\frac{-(1)}{((1 - (d)^{2})^{\frac{1}{2}})})\\=&\frac{arccos(d)}{d} - \frac{ln(d)}{(-d^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{arccos(d)}{d} - \frac{ln(d)}{(-d^{2} + 1)^{\frac{1}{2}}}\right)}{dd}\\=&\frac{-arccos(d)}{d^{2}} + \frac{(\frac{-(1)}{((1 - (d)^{2})^{\frac{1}{2}})})}{d} - (\frac{\frac{-1}{2}(-2d + 0)}{(-d^{2} + 1)^{\frac{3}{2}}})ln(d) - \frac{1}{(-d^{2} + 1)^{\frac{1}{2}}(d)}\\=&\frac{-arccos(d)}{d^{2}} - \frac{dln(d)}{(-d^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-d^{2} + 1)^{\frac{1}{2}}d} - \frac{1}{(-d^{2} + 1)^{\frac{1}{2}}d}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-arccos(d)}{d^{2}} - \frac{dln(d)}{(-d^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-d^{2} + 1)^{\frac{1}{2}}d} - \frac{1}{(-d^{2} + 1)^{\frac{1}{2}}d}\right)}{dd}\\=&\frac{--2arccos(d)}{d^{3}} - \frac{(\frac{-(1)}{((1 - (d)^{2})^{\frac{1}{2}})})}{d^{2}} - (\frac{\frac{-3}{2}(-2d + 0)}{(-d^{2} + 1)^{\frac{5}{2}}})dln(d) - \frac{ln(d)}{(-d^{2} + 1)^{\frac{3}{2}}} - \frac{d}{(-d^{2} + 1)^{\frac{3}{2}}(d)} - \frac{(\frac{\frac{-1}{2}(-2d + 0)}{(-d^{2} + 1)^{\frac{3}{2}}})}{d} - \frac{-1}{(-d^{2} + 1)^{\frac{1}{2}}d^{2}} - \frac{(\frac{\frac{-1}{2}(-2d + 0)}{(-d^{2} + 1)^{\frac{3}{2}}})}{d} - \frac{-1}{(-d^{2} + 1)^{\frac{1}{2}}d^{2}}\\=&\frac{2arccos(d)}{d^{3}} - \frac{3d^{2}ln(d)}{(-d^{2} + 1)^{\frac{5}{2}}} - \frac{ln(d)}{(-d^{2} + 1)^{\frac{3}{2}}} + \frac{1}{(-d^{2} + 1)^{\frac{1}{2}}d^{2}} + \frac{2}{(-d^{2} + 1)^{\frac{1}{2}}d^{2}} - \frac{3}{(-d^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2arccos(d)}{d^{3}} - \frac{3d^{2}ln(d)}{(-d^{2} + 1)^{\frac{5}{2}}} - \frac{ln(d)}{(-d^{2} + 1)^{\frac{3}{2}}} + \frac{1}{(-d^{2} + 1)^{\frac{1}{2}}d^{2}} + \frac{2}{(-d^{2} + 1)^{\frac{1}{2}}d^{2}} - \frac{3}{(-d^{2} + 1)^{\frac{3}{2}}}\right)}{dd}\\=&\frac{2*-3arccos(d)}{d^{4}} + \frac{2(\frac{-(1)}{((1 - (d)^{2})^{\frac{1}{2}})})}{d^{3}} - 3(\frac{\frac{-5}{2}(-2d + 0)}{(-d^{2} + 1)^{\frac{7}{2}}})d^{2}ln(d) - \frac{3*2dln(d)}{(-d^{2} + 1)^{\frac{5}{2}}} - \frac{3d^{2}}{(-d^{2} + 1)^{\frac{5}{2}}(d)} - (\frac{\frac{-3}{2}(-2d + 0)}{(-d^{2} + 1)^{\frac{5}{2}}})ln(d) - \frac{1}{(-d^{2} + 1)^{\frac{3}{2}}(d)} + \frac{(\frac{\frac{-1}{2}(-2d + 0)}{(-d^{2} + 1)^{\frac{3}{2}}})}{d^{2}} + \frac{-2}{(-d^{2} + 1)^{\frac{1}{2}}d^{3}} + \frac{2(\frac{\frac{-1}{2}(-2d + 0)}{(-d^{2} + 1)^{\frac{3}{2}}})}{d^{2}} + \frac{2*-2}{(-d^{2} + 1)^{\frac{1}{2}}d^{3}} - 3(\frac{\frac{-3}{2}(-2d + 0)}{(-d^{2} + 1)^{\frac{5}{2}}})\\=&\frac{-6arccos(d)}{d^{4}} - \frac{15d^{3}ln(d)}{(-d^{2} + 1)^{\frac{7}{2}}} - \frac{9dln(d)}{(-d^{2} + 1)^{\frac{5}{2}}} - \frac{12d}{(-d^{2} + 1)^{\frac{5}{2}}} - \frac{2}{(-d^{2} + 1)^{\frac{1}{2}}d^{3}} + \frac{2}{(-d^{2} + 1)^{\frac{3}{2}}d} - \frac{6}{(-d^{2} + 1)^{\frac{1}{2}}d^{3}}\\ \end{split}\end{equation} \]

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