There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{(1 + \frac{ln(x)}{2})}{sqrt(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{1}{sqrt(x)} + \frac{\frac{1}{2}ln(x)}{sqrt(x)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{1}{sqrt(x)} + \frac{\frac{1}{2}ln(x)}{sqrt(x)}\right)}{dx}\\=&\frac{-\frac{1}{2}}{(x)(x)^{\frac{1}{2}}} + \frac{\frac{1}{2}}{(x)sqrt(x)} + \frac{\frac{1}{2}ln(x)*-\frac{1}{2}}{(x)(x)^{\frac{1}{2}}}\\=& - \frac{ln(x)}{4x^{\frac{3}{2}}} + \frac{1}{2xsqrt(x)} - \frac{1}{2x^{\frac{3}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( - \frac{ln(x)}{4x^{\frac{3}{2}}} + \frac{1}{2xsqrt(x)} - \frac{1}{2x^{\frac{3}{2}}}\right)}{dx}\\=& - \frac{\frac{-3}{2}ln(x)}{4x^{\frac{5}{2}}} - \frac{1}{4x^{\frac{3}{2}}(x)} + \frac{-1}{2x^{2}sqrt(x)} + \frac{-\frac{1}{2}}{2x(x)(x)^{\frac{1}{2}}} - \frac{\frac{-3}{2}}{2x^{\frac{5}{2}}}\\=&\frac{3ln(x)}{8x^{\frac{5}{2}}} - \frac{1}{2x^{2}sqrt(x)} + \frac{1}{4x^{\frac{5}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{3ln(x)}{8x^{\frac{5}{2}}} - \frac{1}{2x^{2}sqrt(x)} + \frac{1}{4x^{\frac{5}{2}}}\right)}{dx}\\=&\frac{3*\frac{-5}{2}ln(x)}{8x^{\frac{7}{2}}} + \frac{3}{8x^{\frac{5}{2}}(x)} - \frac{-2}{2x^{3}sqrt(x)} - \frac{-\frac{1}{2}}{2x^{2}(x)(x)^{\frac{1}{2}}} + \frac{\frac{-5}{2}}{4x^{\frac{7}{2}}}\\=& - \frac{15ln(x)}{16x^{\frac{7}{2}}} + \frac{1}{x^{3}sqrt(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - \frac{15ln(x)}{16x^{\frac{7}{2}}} + \frac{1}{x^{3}sqrt(x)}\right)}{dx}\\=& - \frac{15*\frac{-7}{2}ln(x)}{16x^{\frac{9}{2}}} - \frac{15}{16x^{\frac{7}{2}}(x)} + \frac{-3}{x^{4}sqrt(x)} + \frac{-\frac{1}{2}}{x^{3}(x)(x)^{\frac{1}{2}}}\\=&\frac{105ln(x)}{32x^{\frac{9}{2}}} - \frac{3}{x^{4}sqrt(x)} - \frac{23}{16x^{\frac{9}{2}}}\\ \end{split}\end{equation} \]

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