There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{cos({x}^{2}){\frac{1}{(1 + {x}^{2})}}^{1}}{2}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{\frac{1}{2}cos(x^{2})}{(x^{2} + 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{\frac{1}{2}cos(x^{2})}{(x^{2} + 1)}\right)}{dx}\\=&\frac{1}{2}(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})cos(x^{2}) + \frac{\frac{1}{2}*-sin(x^{2})*2x}{(x^{2} + 1)}\\=&\frac{-xcos(x^{2})}{(x^{2} + 1)^{2}} - \frac{xsin(x^{2})}{(x^{2} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-xcos(x^{2})}{(x^{2} + 1)^{2}} - \frac{xsin(x^{2})}{(x^{2} + 1)}\right)}{dx}\\=&-(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})xcos(x^{2}) - \frac{cos(x^{2})}{(x^{2} + 1)^{2}} - \frac{x*-sin(x^{2})*2x}{(x^{2} + 1)^{2}} - (\frac{-(2x + 0)}{(x^{2} + 1)^{2}})xsin(x^{2}) - \frac{sin(x^{2})}{(x^{2} + 1)} - \frac{xcos(x^{2})*2x}{(x^{2} + 1)}\\=&\frac{4x^{2}cos(x^{2})}{(x^{2} + 1)^{3}} - \frac{cos(x^{2})}{(x^{2} + 1)^{2}} + \frac{4x^{2}sin(x^{2})}{(x^{2} + 1)^{2}} - \frac{sin(x^{2})}{(x^{2} + 1)} - \frac{2x^{2}cos(x^{2})}{(x^{2} + 1)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{4x^{2}cos(x^{2})}{(x^{2} + 1)^{3}} - \frac{cos(x^{2})}{(x^{2} + 1)^{2}} + \frac{4x^{2}sin(x^{2})}{(x^{2} + 1)^{2}} - \frac{sin(x^{2})}{(x^{2} + 1)} - \frac{2x^{2}cos(x^{2})}{(x^{2} + 1)}\right)}{dx}\\=&4(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2}cos(x^{2}) + \frac{4*2xcos(x^{2})}{(x^{2} + 1)^{3}} + \frac{4x^{2}*-sin(x^{2})*2x}{(x^{2} + 1)^{3}} - (\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})cos(x^{2}) - \frac{-sin(x^{2})*2x}{(x^{2} + 1)^{2}} + 4(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{2}sin(x^{2}) + \frac{4*2xsin(x^{2})}{(x^{2} + 1)^{2}} + \frac{4x^{2}cos(x^{2})*2x}{(x^{2} + 1)^{2}} - (\frac{-(2x + 0)}{(x^{2} + 1)^{2}})sin(x^{2}) - \frac{cos(x^{2})*2x}{(x^{2} + 1)} - 2(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x^{2}cos(x^{2}) - \frac{2*2xcos(x^{2})}{(x^{2} + 1)} - \frac{2x^{2}*-sin(x^{2})*2x}{(x^{2} + 1)}\\=&\frac{-24x^{3}cos(x^{2})}{(x^{2} + 1)^{4}} + \frac{12xcos(x^{2})}{(x^{2} + 1)^{3}} - \frac{24x^{3}sin(x^{2})}{(x^{2} + 1)^{3}} + \frac{12xsin(x^{2})}{(x^{2} + 1)^{2}} + \frac{12x^{3}cos(x^{2})}{(x^{2} + 1)^{2}} - \frac{6xcos(x^{2})}{(x^{2} + 1)} + \frac{4x^{3}sin(x^{2})}{(x^{2} + 1)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-24x^{3}cos(x^{2})}{(x^{2} + 1)^{4}} + \frac{12xcos(x^{2})}{(x^{2} + 1)^{3}} - \frac{24x^{3}sin(x^{2})}{(x^{2} + 1)^{3}} + \frac{12xsin(x^{2})}{(x^{2} + 1)^{2}} + \frac{12x^{3}cos(x^{2})}{(x^{2} + 1)^{2}} - \frac{6xcos(x^{2})}{(x^{2} + 1)} + \frac{4x^{3}sin(x^{2})}{(x^{2} + 1)}\right)}{dx}\\=&-24(\frac{-4(2x + 0)}{(x^{2} + 1)^{5}})x^{3}cos(x^{2}) - \frac{24*3x^{2}cos(x^{2})}{(x^{2} + 1)^{4}} - \frac{24x^{3}*-sin(x^{2})*2x}{(x^{2} + 1)^{4}} + 12(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})xcos(x^{2}) + \frac{12cos(x^{2})}{(x^{2} + 1)^{3}} + \frac{12x*-sin(x^{2})*2x}{(x^{2} + 1)^{3}} - 24(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{3}sin(x^{2}) - \frac{24*3x^{2}sin(x^{2})}{(x^{2} + 1)^{3}} - \frac{24x^{3}cos(x^{2})*2x}{(x^{2} + 1)^{3}} + 12(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})xsin(x^{2}) + \frac{12sin(x^{2})}{(x^{2} + 1)^{2}} + \frac{12xcos(x^{2})*2x}{(x^{2} + 1)^{2}} + 12(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{3}cos(x^{2}) + \frac{12*3x^{2}cos(x^{2})}{(x^{2} + 1)^{2}} + \frac{12x^{3}*-sin(x^{2})*2x}{(x^{2} + 1)^{2}} - 6(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})xcos(x^{2}) - \frac{6cos(x^{2})}{(x^{2} + 1)} - \frac{6x*-sin(x^{2})*2x}{(x^{2} + 1)} + 4(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x^{3}sin(x^{2}) + \frac{4*3x^{2}sin(x^{2})}{(x^{2} + 1)} + \frac{4x^{3}cos(x^{2})*2x}{(x^{2} + 1)}\\=&\frac{192x^{4}cos(x^{2})}{(x^{2} + 1)^{5}} - \frac{144x^{2}cos(x^{2})}{(x^{2} + 1)^{4}} + \frac{192x^{4}sin(x^{2})}{(x^{2} + 1)^{4}} + \frac{12cos(x^{2})}{(x^{2} + 1)^{3}} - \frac{144x^{2}sin(x^{2})}{(x^{2} + 1)^{3}} - \frac{96x^{4}cos(x^{2})}{(x^{2} + 1)^{3}} + \frac{12sin(x^{2})}{(x^{2} + 1)^{2}} + \frac{72x^{2}cos(x^{2})}{(x^{2} + 1)^{2}} - \frac{32x^{4}sin(x^{2})}{(x^{2} + 1)^{2}} - \frac{6cos(x^{2})}{(x^{2} + 1)} + \frac{24x^{2}sin(x^{2})}{(x^{2} + 1)} + \frac{8x^{4}cos(x^{2})}{(x^{2} + 1)}\\ \end{split}\end{equation} \]

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