There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {(1 + {x}^{2})}^{\frac{1}{2}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = (x^{2} + 1)^{\frac{1}{2}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( (x^{2} + 1)^{\frac{1}{2}}\right)}{dx}\\=&(\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}})\\=&\frac{x}{(x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{x}{(x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})x + \frac{1}{(x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-x^{2}}{(x^{2} + 1)^{\frac{3}{2}}} + \frac{1}{(x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-x^{2}}{(x^{2} + 1)^{\frac{3}{2}}} + \frac{1}{(x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x^{2} - \frac{2x}{(x^{2} + 1)^{\frac{3}{2}}} + (\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})\\=&\frac{3x^{3}}{(x^{2} + 1)^{\frac{5}{2}}} - \frac{3x}{(x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3x^{3}}{(x^{2} + 1)^{\frac{5}{2}}} - \frac{3x}{(x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(2x + 0)}{(x^{2} + 1)^{\frac{7}{2}}})x^{3} + \frac{3*3x^{2}}{(x^{2} + 1)^{\frac{5}{2}}} - 3(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x - \frac{3}{(x^{2} + 1)^{\frac{3}{2}}}\\=&\frac{-15x^{4}}{(x^{2} + 1)^{\frac{7}{2}}} + \frac{18x^{2}}{(x^{2} + 1)^{\frac{5}{2}}} - \frac{3}{(x^{2} + 1)^{\frac{3}{2}}}\\ \end{split}\end{equation} \]

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