There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ x{\frac{1}{(1 - {x}^{2})}}^{(\frac{3}{2})} + \frac{2x}{(1 + {x}^{2})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{2x}{(x^{2} + 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{2x}{(x^{2} + 1)}\right)}{dx}\\=&(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}} + 2(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x + \frac{2}{(x^{2} + 1)}\\=&\frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{4x^{2}}{(x^{2} + 1)^{2}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{2}{(x^{2} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{4x^{2}}{(x^{2} + 1)^{2}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{2}{(x^{2} + 1)}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} + \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} - 4(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{2} - \frac{4*2x}{(x^{2} + 1)^{2}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}}) + 2(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})\\=&\frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{16x^{3}}{(x^{2} + 1)^{3}} - \frac{12x}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{16x^{3}}{(x^{2} + 1)^{3}} - \frac{12x}{(x^{2} + 1)^{2}}\right)}{dx}\\=&15(\frac{\frac{-7}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{9}{2}}})x^{3} + \frac{15*3x^{2}}{(-x^{2} + 1)^{\frac{7}{2}}} + 9(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x + \frac{9}{(-x^{2} + 1)^{\frac{5}{2}}} + 16(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{3} + \frac{16*3x^{2}}{(x^{2} + 1)^{3}} - 12(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x - \frac{12}{(x^{2} + 1)^{2}}\\=&\frac{105x^{4}}{(-x^{2} + 1)^{\frac{9}{2}}} + \frac{90x^{2}}{(-x^{2} + 1)^{\frac{7}{2}}} - \frac{96x^{4}}{(x^{2} + 1)^{4}} + \frac{96x^{2}}{(x^{2} + 1)^{3}} - \frac{12}{(x^{2} + 1)^{2}} + \frac{9}{(-x^{2} + 1)^{\frac{5}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{105x^{4}}{(-x^{2} + 1)^{\frac{9}{2}}} + \frac{90x^{2}}{(-x^{2} + 1)^{\frac{7}{2}}} - \frac{96x^{4}}{(x^{2} + 1)^{4}} + \frac{96x^{2}}{(x^{2} + 1)^{3}} - \frac{12}{(x^{2} + 1)^{2}} + \frac{9}{(-x^{2} + 1)^{\frac{5}{2}}}\right)}{dx}\\=&105(\frac{\frac{-9}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{11}{2}}})x^{4} + \frac{105*4x^{3}}{(-x^{2} + 1)^{\frac{9}{2}}} + 90(\frac{\frac{-7}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{9}{2}}})x^{2} + \frac{90*2x}{(-x^{2} + 1)^{\frac{7}{2}}} - 96(\frac{-4(2x + 0)}{(x^{2} + 1)^{5}})x^{4} - \frac{96*4x^{3}}{(x^{2} + 1)^{4}} + 96(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2} + \frac{96*2x}{(x^{2} + 1)^{3}} - 12(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}}) + 9(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})\\=&\frac{945x^{5}}{(-x^{2} + 1)^{\frac{11}{2}}} + \frac{1050x^{3}}{(-x^{2} + 1)^{\frac{9}{2}}} + \frac{225x}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{768x^{5}}{(x^{2} + 1)^{5}} - \frac{960x^{3}}{(x^{2} + 1)^{4}} + \frac{240x}{(x^{2} + 1)^{3}}\\ \end{split}\end{equation} \]

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