There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ e^{x} + log_{x}^{sin(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{x} + log_{x}^{sin(x)}\right)}{dx}\\=&e^{x} + (\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})\\=&e^{x} + \frac{cos(x)}{ln(x)sin(x)} - \frac{log_{x}^{sin(x)}}{xln(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( e^{x} + \frac{cos(x)}{ln(x)sin(x)} - \frac{log_{x}^{sin(x)}}{xln(x)}\right)}{dx}\\=&e^{x} + \frac{-cos(x)}{ln^{2}(x)(x)sin(x)} + \frac{-cos(x)cos(x)}{ln(x)sin^{2}(x)} + \frac{-sin(x)}{ln(x)sin(x)} - \frac{-log_{x}^{sin(x)}}{x^{2}ln(x)} - \frac{(\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})}{xln(x)} - \frac{log_{x}^{sin(x)}*-1}{xln^{2}(x)(x)}\\=&e^{x} - \frac{2cos(x)}{xln^{2}(x)sin(x)} - \frac{cos^{2}(x)}{ln(x)sin^{2}(x)} - \frac{1}{ln(x)} + \frac{log_{x}^{sin(x)}}{x^{2}ln(x)} + \frac{2log_{x}^{sin(x)}}{x^{2}ln^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( e^{x} - \frac{2cos(x)}{xln^{2}(x)sin(x)} - \frac{cos^{2}(x)}{ln(x)sin^{2}(x)} - \frac{1}{ln(x)} + \frac{log_{x}^{sin(x)}}{x^{2}ln(x)} + \frac{2log_{x}^{sin(x)}}{x^{2}ln^{2}(x)}\right)}{dx}\\=&e^{x} - \frac{2*-cos(x)}{x^{2}ln^{2}(x)sin(x)} - \frac{2*-2cos(x)}{xln^{3}(x)(x)sin(x)} - \frac{2*-cos(x)cos(x)}{xln^{2}(x)sin^{2}(x)} - \frac{2*-sin(x)}{xln^{2}(x)sin(x)} - \frac{-cos^{2}(x)}{ln^{2}(x)(x)sin^{2}(x)} - \frac{-2cos(x)cos^{2}(x)}{ln(x)sin^{3}(x)} - \frac{-2cos(x)sin(x)}{ln(x)sin^{2}(x)} - \frac{-1}{ln^{2}(x)(x)} + \frac{-2log_{x}^{sin(x)}}{x^{3}ln(x)} + \frac{(\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})}{x^{2}ln(x)} + \frac{log_{x}^{sin(x)}*-1}{x^{2}ln^{2}(x)(x)} + \frac{2*-2log_{x}^{sin(x)}}{x^{3}ln^{2}(x)} + \frac{2(\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})}{x^{2}ln^{2}(x)} + \frac{2log_{x}^{sin(x)}*-2}{x^{2}ln^{3}(x)(x)}\\=&e^{x} + \frac{3cos(x)}{x^{2}ln^{2}(x)sin(x)} + \frac{6cos(x)}{x^{2}ln^{3}(x)sin(x)} + \frac{3cos^{2}(x)}{xln^{2}(x)sin^{2}(x)} + \frac{3}{xln^{2}(x)} + \frac{2cos^{3}(x)}{ln(x)sin^{3}(x)} + \frac{2cos(x)}{ln(x)sin(x)} - \frac{2log_{x}^{sin(x)}}{x^{3}ln(x)} - \frac{6log_{x}^{sin(x)}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{sin(x)}}{x^{3}ln^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( e^{x} + \frac{3cos(x)}{x^{2}ln^{2}(x)sin(x)} + \frac{6cos(x)}{x^{2}ln^{3}(x)sin(x)} + \frac{3cos^{2}(x)}{xln^{2}(x)sin^{2}(x)} + \frac{3}{xln^{2}(x)} + \frac{2cos^{3}(x)}{ln(x)sin^{3}(x)} + \frac{2cos(x)}{ln(x)sin(x)} - \frac{2log_{x}^{sin(x)}}{x^{3}ln(x)} - \frac{6log_{x}^{sin(x)}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{sin(x)}}{x^{3}ln^{3}(x)}\right)}{dx}\\=&e^{x} + \frac{3*-2cos(x)}{x^{3}ln^{2}(x)sin(x)} + \frac{3*-2cos(x)}{x^{2}ln^{3}(x)(x)sin(x)} + \frac{3*-cos(x)cos(x)}{x^{2}ln^{2}(x)sin^{2}(x)} + \frac{3*-sin(x)}{x^{2}ln^{2}(x)sin(x)} + \frac{6*-2cos(x)}{x^{3}ln^{3}(x)sin(x)} + \frac{6*-3cos(x)}{x^{2}ln^{4}(x)(x)sin(x)} + \frac{6*-cos(x)cos(x)}{x^{2}ln^{3}(x)sin^{2}(x)} + \frac{6*-sin(x)}{x^{2}ln^{3}(x)sin(x)} + \frac{3*-cos^{2}(x)}{x^{2}ln^{2}(x)sin^{2}(x)} + \frac{3*-2cos^{2}(x)}{xln^{3}(x)(x)sin^{2}(x)} + \frac{3*-2cos(x)cos^{2}(x)}{xln^{2}(x)sin^{3}(x)} + \frac{3*-2cos(x)sin(x)}{xln^{2}(x)sin^{2}(x)} + \frac{3*-1}{x^{2}ln^{2}(x)} + \frac{3*-2}{xln^{3}(x)(x)} + \frac{2*-cos^{3}(x)}{ln^{2}(x)(x)sin^{3}(x)} + \frac{2*-3cos(x)cos^{3}(x)}{ln(x)sin^{4}(x)} + \frac{2*-3cos^{2}(x)sin(x)}{ln(x)sin^{3}(x)} + \frac{2*-cos(x)}{ln^{2}(x)(x)sin(x)} + \frac{2*-cos(x)cos(x)}{ln(x)sin^{2}(x)} + \frac{2*-sin(x)}{ln(x)sin(x)} - \frac{2*-3log_{x}^{sin(x)}}{x^{4}ln(x)} - \frac{2(\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})}{x^{3}ln(x)} - \frac{2log_{x}^{sin(x)}*-1}{x^{3}ln^{2}(x)(x)} - \frac{6*-3log_{x}^{sin(x)}}{x^{4}ln^{2}(x)} - \frac{6(\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{sin(x)}*-2}{x^{3}ln^{3}(x)(x)} - \frac{6*-3log_{x}^{sin(x)}}{x^{4}ln^{3}(x)} - \frac{6(\frac{(\frac{(cos(x))}{(sin(x))} - \frac{(1)log_{x}^{sin(x)}}{(x)})}{(ln(x))})}{x^{3}ln^{3}(x)} - \frac{6log_{x}^{sin(x)}*-3}{x^{3}ln^{4}(x)(x)}\\=&e^{x} - \frac{8cos(x)}{x^{3}ln^{2}(x)sin(x)} - \frac{24cos(x)}{x^{3}ln^{3}(x)sin(x)} - \frac{6cos^{2}(x)}{x^{2}ln^{2}(x)sin^{2}(x)} - \frac{8cos(x)}{xln^{2}(x)sin(x)} - \frac{24cos(x)}{x^{3}ln^{4}(x)sin(x)} - \frac{12cos^{2}(x)}{x^{2}ln^{3}(x)sin^{2}(x)} - \frac{8cos^{3}(x)}{xln^{2}(x)sin^{3}(x)} - \frac{12}{x^{2}ln^{3}(x)} - \frac{6}{x^{2}ln^{2}(x)} - \frac{6cos^{4}(x)}{ln(x)sin^{4}(x)} - \frac{8cos^{2}(x)}{ln(x)sin^{2}(x)} - \frac{2}{ln(x)} + \frac{6log_{x}^{sin(x)}}{x^{4}ln(x)} + \frac{22log_{x}^{sin(x)}}{x^{4}ln^{2}(x)} + \frac{36log_{x}^{sin(x)}}{x^{4}ln^{3}(x)} + \frac{24log_{x}^{sin(x)}}{x^{4}ln^{4}(x)}\\ \end{split}\end{equation} \]

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