There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ cos({x}^{2}){\frac{1}{(1 + {x}^{2})}}^{\frac{1}{2}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{cos(x^{2})}{(x^{2} + 1)^{\frac{1}{2}}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{cos(x^{2})}{(x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})cos(x^{2}) + \frac{-sin(x^{2})*2x}{(x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-xcos(x^{2})}{(x^{2} + 1)^{\frac{3}{2}}} - \frac{2xsin(x^{2})}{(x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-xcos(x^{2})}{(x^{2} + 1)^{\frac{3}{2}}} - \frac{2xsin(x^{2})}{(x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})xcos(x^{2}) - \frac{cos(x^{2})}{(x^{2} + 1)^{\frac{3}{2}}} - \frac{x*-sin(x^{2})*2x}{(x^{2} + 1)^{\frac{3}{2}}} - 2(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})xsin(x^{2}) - \frac{2sin(x^{2})}{(x^{2} + 1)^{\frac{1}{2}}} - \frac{2xcos(x^{2})*2x}{(x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{3x^{2}cos(x^{2})}{(x^{2} + 1)^{\frac{5}{2}}} - \frac{cos(x^{2})}{(x^{2} + 1)^{\frac{3}{2}}} + \frac{4x^{2}sin(x^{2})}{(x^{2} + 1)^{\frac{3}{2}}} - \frac{2sin(x^{2})}{(x^{2} + 1)^{\frac{1}{2}}} - \frac{4x^{2}cos(x^{2})}{(x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{3x^{2}cos(x^{2})}{(x^{2} + 1)^{\frac{5}{2}}} - \frac{cos(x^{2})}{(x^{2} + 1)^{\frac{3}{2}}} + \frac{4x^{2}sin(x^{2})}{(x^{2} + 1)^{\frac{3}{2}}} - \frac{2sin(x^{2})}{(x^{2} + 1)^{\frac{1}{2}}} - \frac{4x^{2}cos(x^{2})}{(x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(2x + 0)}{(x^{2} + 1)^{\frac{7}{2}}})x^{2}cos(x^{2}) + \frac{3*2xcos(x^{2})}{(x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}*-sin(x^{2})*2x}{(x^{2} + 1)^{\frac{5}{2}}} - (\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})cos(x^{2}) - \frac{-sin(x^{2})*2x}{(x^{2} + 1)^{\frac{3}{2}}} + 4(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x^{2}sin(x^{2}) + \frac{4*2xsin(x^{2})}{(x^{2} + 1)^{\frac{3}{2}}} + \frac{4x^{2}cos(x^{2})*2x}{(x^{2} + 1)^{\frac{3}{2}}} - 2(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})sin(x^{2}) - \frac{2cos(x^{2})*2x}{(x^{2} + 1)^{\frac{1}{2}}} - 4(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})x^{2}cos(x^{2}) - \frac{4*2xcos(x^{2})}{(x^{2} + 1)^{\frac{1}{2}}} - \frac{4x^{2}*-sin(x^{2})*2x}{(x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-15x^{3}cos(x^{2})}{(x^{2} + 1)^{\frac{7}{2}}} + \frac{9xcos(x^{2})}{(x^{2} + 1)^{\frac{5}{2}}} - \frac{18x^{3}sin(x^{2})}{(x^{2} + 1)^{\frac{5}{2}}} + \frac{12xsin(x^{2})}{(x^{2} + 1)^{\frac{3}{2}}} + \frac{12x^{3}cos(x^{2})}{(x^{2} + 1)^{\frac{3}{2}}} - \frac{12xcos(x^{2})}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{8x^{3}sin(x^{2})}{(x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-15x^{3}cos(x^{2})}{(x^{2} + 1)^{\frac{7}{2}}} + \frac{9xcos(x^{2})}{(x^{2} + 1)^{\frac{5}{2}}} - \frac{18x^{3}sin(x^{2})}{(x^{2} + 1)^{\frac{5}{2}}} + \frac{12xsin(x^{2})}{(x^{2} + 1)^{\frac{3}{2}}} + \frac{12x^{3}cos(x^{2})}{(x^{2} + 1)^{\frac{3}{2}}} - \frac{12xcos(x^{2})}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{8x^{3}sin(x^{2})}{(x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-15(\frac{\frac{-7}{2}(2x + 0)}{(x^{2} + 1)^{\frac{9}{2}}})x^{3}cos(x^{2}) - \frac{15*3x^{2}cos(x^{2})}{(x^{2} + 1)^{\frac{7}{2}}} - \frac{15x^{3}*-sin(x^{2})*2x}{(x^{2} + 1)^{\frac{7}{2}}} + 9(\frac{\frac{-5}{2}(2x + 0)}{(x^{2} + 1)^{\frac{7}{2}}})xcos(x^{2}) + \frac{9cos(x^{2})}{(x^{2} + 1)^{\frac{5}{2}}} + \frac{9x*-sin(x^{2})*2x}{(x^{2} + 1)^{\frac{5}{2}}} - 18(\frac{\frac{-5}{2}(2x + 0)}{(x^{2} + 1)^{\frac{7}{2}}})x^{3}sin(x^{2}) - \frac{18*3x^{2}sin(x^{2})}{(x^{2} + 1)^{\frac{5}{2}}} - \frac{18x^{3}cos(x^{2})*2x}{(x^{2} + 1)^{\frac{5}{2}}} + 12(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})xsin(x^{2}) + \frac{12sin(x^{2})}{(x^{2} + 1)^{\frac{3}{2}}} + \frac{12xcos(x^{2})*2x}{(x^{2} + 1)^{\frac{3}{2}}} + 12(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x^{3}cos(x^{2}) + \frac{12*3x^{2}cos(x^{2})}{(x^{2} + 1)^{\frac{3}{2}}} + \frac{12x^{3}*-sin(x^{2})*2x}{(x^{2} + 1)^{\frac{3}{2}}} - 12(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})xcos(x^{2}) - \frac{12cos(x^{2})}{(x^{2} + 1)^{\frac{1}{2}}} - \frac{12x*-sin(x^{2})*2x}{(x^{2} + 1)^{\frac{1}{2}}} + 8(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})x^{3}sin(x^{2}) + \frac{8*3x^{2}sin(x^{2})}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{8x^{3}cos(x^{2})*2x}{(x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{105x^{4}cos(x^{2})}{(x^{2} + 1)^{\frac{9}{2}}} - \frac{90x^{2}cos(x^{2})}{(x^{2} + 1)^{\frac{7}{2}}} + \frac{120x^{4}sin(x^{2})}{(x^{2} + 1)^{\frac{7}{2}}} + \frac{9cos(x^{2})}{(x^{2} + 1)^{\frac{5}{2}}} - \frac{108x^{2}sin(x^{2})}{(x^{2} + 1)^{\frac{5}{2}}} - \frac{72x^{4}cos(x^{2})}{(x^{2} + 1)^{\frac{5}{2}}} + \frac{12sin(x^{2})}{(x^{2} + 1)^{\frac{3}{2}}} + \frac{72x^{2}cos(x^{2})}{(x^{2} + 1)^{\frac{3}{2}}} - \frac{32x^{4}sin(x^{2})}{(x^{2} + 1)^{\frac{3}{2}}} - \frac{12cos(x^{2})}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{48x^{2}sin(x^{2})}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{16x^{4}cos(x^{2})}{(x^{2} + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]

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