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

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