There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ xtan(2{x}^{\frac{1}{2}}) + x\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = xtan(2x^{\frac{1}{2}}) + x\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( xtan(2x^{\frac{1}{2}}) + x\right)}{dx}\\=&tan(2x^{\frac{1}{2}}) + xsec^{2}(2x^{\frac{1}{2}})(\frac{2*\frac{1}{2}}{x^{\frac{1}{2}}}) + 1\\=&tan(2x^{\frac{1}{2}}) + x^{\frac{1}{2}}sec^{2}(2x^{\frac{1}{2}}) + 1\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( tan(2x^{\frac{1}{2}}) + x^{\frac{1}{2}}sec^{2}(2x^{\frac{1}{2}}) + 1\right)}{dx}\\=&sec^{2}(2x^{\frac{1}{2}})(\frac{2*\frac{1}{2}}{x^{\frac{1}{2}}}) + \frac{\frac{1}{2}sec^{2}(2x^{\frac{1}{2}})}{x^{\frac{1}{2}}} + \frac{x^{\frac{1}{2}}*2sec^{2}(2x^{\frac{1}{2}})tan(2x^{\frac{1}{2}})*2*\frac{1}{2}}{x^{\frac{1}{2}}} + 0\\=&\frac{3sec^{2}(2x^{\frac{1}{2}})}{2x^{\frac{1}{2}}} + 2tan(2x^{\frac{1}{2}})sec^{2}(2x^{\frac{1}{2}})\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{3sec^{2}(2x^{\frac{1}{2}})}{2x^{\frac{1}{2}}} + 2tan(2x^{\frac{1}{2}})sec^{2}(2x^{\frac{1}{2}})\right)}{dx}\\=&\frac{3*\frac{-1}{2}sec^{2}(2x^{\frac{1}{2}})}{2x^{\frac{3}{2}}} + \frac{3*2sec^{2}(2x^{\frac{1}{2}})tan(2x^{\frac{1}{2}})*2*\frac{1}{2}}{2x^{\frac{1}{2}}x^{\frac{1}{2}}} + 2sec^{2}(2x^{\frac{1}{2}})(\frac{2*\frac{1}{2}}{x^{\frac{1}{2}}})sec^{2}(2x^{\frac{1}{2}}) + \frac{2tan(2x^{\frac{1}{2}})*2sec^{2}(2x^{\frac{1}{2}})tan(2x^{\frac{1}{2}})*2*\frac{1}{2}}{x^{\frac{1}{2}}}\\=&\frac{2sec^{4}(2x^{\frac{1}{2}})}{x^{\frac{1}{2}}} + \frac{3tan(2x^{\frac{1}{2}})sec^{2}(2x^{\frac{1}{2}})}{x} - \frac{3sec^{2}(2x^{\frac{1}{2}})}{4x^{\frac{3}{2}}} + \frac{4tan^{2}(2x^{\frac{1}{2}})sec^{2}(2x^{\frac{1}{2}})}{x^{\frac{1}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2sec^{4}(2x^{\frac{1}{2}})}{x^{\frac{1}{2}}} + \frac{3tan(2x^{\frac{1}{2}})sec^{2}(2x^{\frac{1}{2}})}{x} - \frac{3sec^{2}(2x^{\frac{1}{2}})}{4x^{\frac{3}{2}}} + \frac{4tan^{2}(2x^{\frac{1}{2}})sec^{2}(2x^{\frac{1}{2}})}{x^{\frac{1}{2}}}\right)}{dx}\\=&\frac{2*\frac{-1}{2}sec^{4}(2x^{\frac{1}{2}})}{x^{\frac{3}{2}}} + \frac{2*4sec^{4}(2x^{\frac{1}{2}})tan(2x^{\frac{1}{2}})*2*\frac{1}{2}}{x^{\frac{1}{2}}x^{\frac{1}{2}}} + \frac{3*-tan(2x^{\frac{1}{2}})sec^{2}(2x^{\frac{1}{2}})}{x^{2}} + \frac{3sec^{2}(2x^{\frac{1}{2}})(\frac{2*\frac{1}{2}}{x^{\frac{1}{2}}})sec^{2}(2x^{\frac{1}{2}})}{x} + \frac{3tan(2x^{\frac{1}{2}})*2sec^{2}(2x^{\frac{1}{2}})tan(2x^{\frac{1}{2}})*2*\frac{1}{2}}{xx^{\frac{1}{2}}} - \frac{3*\frac{-3}{2}sec^{2}(2x^{\frac{1}{2}})}{4x^{\frac{5}{2}}} - \frac{3*2sec^{2}(2x^{\frac{1}{2}})tan(2x^{\frac{1}{2}})*2*\frac{1}{2}}{4x^{\frac{3}{2}}x^{\frac{1}{2}}} + \frac{4*\frac{-1}{2}tan^{2}(2x^{\frac{1}{2}})sec^{2}(2x^{\frac{1}{2}})}{x^{\frac{3}{2}}} + \frac{4*2tan(2x^{\frac{1}{2}})sec^{2}(2x^{\frac{1}{2}})(\frac{2*\frac{1}{2}}{x^{\frac{1}{2}}})sec^{2}(2x^{\frac{1}{2}})}{x^{\frac{1}{2}}} + \frac{4tan^{2}(2x^{\frac{1}{2}})*2sec^{2}(2x^{\frac{1}{2}})tan(2x^{\frac{1}{2}})*2*\frac{1}{2}}{x^{\frac{1}{2}}x^{\frac{1}{2}}}\\=&\frac{2sec^{4}(2x^{\frac{1}{2}})}{x^{\frac{3}{2}}} + \frac{16tan(2x^{\frac{1}{2}})sec^{4}(2x^{\frac{1}{2}})}{x} - \frac{9tan(2x^{\frac{1}{2}})sec^{2}(2x^{\frac{1}{2}})}{2x^{2}} + \frac{4tan^{2}(2x^{\frac{1}{2}})sec^{2}(2x^{\frac{1}{2}})}{x^{\frac{3}{2}}} + \frac{9sec^{2}(2x^{\frac{1}{2}})}{8x^{\frac{5}{2}}} + \frac{8tan^{3}(2x^{\frac{1}{2}})sec^{2}(2x^{\frac{1}{2}})}{x}\\ \end{split}\end{equation} \]

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