There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ sin(xx + lg(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = sin(x^{2} + lg(x))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin(x^{2} + lg(x))\right)}{dx}\\=&cos(x^{2} + lg(x))(2x + \frac{1}{ln{10}(x)})\\=&2xcos(x^{2} + lg(x)) + \frac{cos(x^{2} + lg(x))}{xln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 2xcos(x^{2} + lg(x)) + \frac{cos(x^{2} + lg(x))}{xln{10}}\right)}{dx}\\=&2cos(x^{2} + lg(x)) + 2x*-sin(x^{2} + lg(x))(2x + \frac{1}{ln{10}(x)}) + \frac{-cos(x^{2} + lg(x))}{x^{2}ln{10}} + \frac{-0cos(x^{2} + lg(x))}{xln^{2}{10}} + \frac{-sin(x^{2} + lg(x))(2x + \frac{1}{ln{10}(x)})}{xln{10}}\\=&2cos(x^{2} + lg(x)) - 4x^{2}sin(x^{2} + lg(x)) - \frac{4sin(x^{2} + lg(x))}{ln{10}} - \frac{cos(x^{2} + lg(x))}{x^{2}ln{10}} - \frac{sin(x^{2} + lg(x))}{x^{2}ln^{2}{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2cos(x^{2} + lg(x)) - 4x^{2}sin(x^{2} + lg(x)) - \frac{4sin(x^{2} + lg(x))}{ln{10}} - \frac{cos(x^{2} + lg(x))}{x^{2}ln{10}} - \frac{sin(x^{2} + lg(x))}{x^{2}ln^{2}{10}}\right)}{dx}\\=&2*-sin(x^{2} + lg(x))(2x + \frac{1}{ln{10}(x)}) - 4*2xsin(x^{2} + lg(x)) - 4x^{2}cos(x^{2} + lg(x))(2x + \frac{1}{ln{10}(x)}) - \frac{4*-0sin(x^{2} + lg(x))}{ln^{2}{10}} - \frac{4cos(x^{2} + lg(x))(2x + \frac{1}{ln{10}(x)})}{ln{10}} - \frac{-2cos(x^{2} + lg(x))}{x^{3}ln{10}} - \frac{-0cos(x^{2} + lg(x))}{x^{2}ln^{2}{10}} - \frac{-sin(x^{2} + lg(x))(2x + \frac{1}{ln{10}(x)})}{x^{2}ln{10}} - \frac{-2sin(x^{2} + lg(x))}{x^{3}ln^{2}{10}} - \frac{-2*0sin(x^{2} + lg(x))}{x^{2}ln^{3}{10}} - \frac{cos(x^{2} + lg(x))(2x + \frac{1}{ln{10}(x)})}{x^{2}ln^{2}{10}}\\=&-12xsin(x^{2} + lg(x)) - 8x^{3}cos(x^{2} + lg(x)) - \frac{12xcos(x^{2} + lg(x))}{ln{10}} - \frac{6cos(x^{2} + lg(x))}{xln^{2}{10}} + \frac{2cos(x^{2} + lg(x))}{x^{3}ln{10}} + \frac{3sin(x^{2} + lg(x))}{x^{3}ln^{2}{10}} - \frac{cos(x^{2} + lg(x))}{x^{3}ln^{3}{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -12xsin(x^{2} + lg(x)) - 8x^{3}cos(x^{2} + lg(x)) - \frac{12xcos(x^{2} + lg(x))}{ln{10}} - \frac{6cos(x^{2} + lg(x))}{xln^{2}{10}} + \frac{2cos(x^{2} + lg(x))}{x^{3}ln{10}} + \frac{3sin(x^{2} + lg(x))}{x^{3}ln^{2}{10}} - \frac{cos(x^{2} + lg(x))}{x^{3}ln^{3}{10}}\right)}{dx}\\=&-12sin(x^{2} + lg(x)) - 12xcos(x^{2} + lg(x))(2x + \frac{1}{ln{10}(x)}) - 8*3x^{2}cos(x^{2} + lg(x)) - 8x^{3}*-sin(x^{2} + lg(x))(2x + \frac{1}{ln{10}(x)}) - \frac{12cos(x^{2} + lg(x))}{ln{10}} - \frac{12x*-0cos(x^{2} + lg(x))}{ln^{2}{10}} - \frac{12x*-sin(x^{2} + lg(x))(2x + \frac{1}{ln{10}(x)})}{ln{10}} - \frac{6*-cos(x^{2} + lg(x))}{x^{2}ln^{2}{10}} - \frac{6*-2*0cos(x^{2} + lg(x))}{xln^{3}{10}} - \frac{6*-sin(x^{2} + lg(x))(2x + \frac{1}{ln{10}(x)})}{xln^{2}{10}} + \frac{2*-3cos(x^{2} + lg(x))}{x^{4}ln{10}} + \frac{2*-0cos(x^{2} + lg(x))}{x^{3}ln^{2}{10}} + \frac{2*-sin(x^{2} + lg(x))(2x + \frac{1}{ln{10}(x)})}{x^{3}ln{10}} + \frac{3*-3sin(x^{2} + lg(x))}{x^{4}ln^{2}{10}} + \frac{3*-2*0sin(x^{2} + lg(x))}{x^{3}ln^{3}{10}} + \frac{3cos(x^{2} + lg(x))(2x + \frac{1}{ln{10}(x)})}{x^{3}ln^{2}{10}} - \frac{-3cos(x^{2} + lg(x))}{x^{4}ln^{3}{10}} - \frac{-3*0cos(x^{2} + lg(x))}{x^{3}ln^{4}{10}} - \frac{-sin(x^{2} + lg(x))(2x + \frac{1}{ln{10}(x)})}{x^{3}ln^{3}{10}}\\=&-12sin(x^{2} + lg(x)) - 48x^{2}cos(x^{2} + lg(x)) - \frac{24cos(x^{2} + lg(x))}{ln{10}} + 16x^{4}sin(x^{2} + lg(x)) + \frac{32x^{2}sin(x^{2} + lg(x))}{ln{10}} + \frac{24sin(x^{2} + lg(x))}{ln^{2}{10}} + \frac{12cos(x^{2} + lg(x))}{x^{2}ln^{2}{10}} + \frac{8sin(x^{2} + lg(x))}{x^{2}ln^{3}{10}} - \frac{6cos(x^{2} + lg(x))}{x^{4}ln{10}} - \frac{4sin(x^{2} + lg(x))}{x^{2}ln{10}} - \frac{11sin(x^{2} + lg(x))}{x^{4}ln^{2}{10}} + \frac{6cos(x^{2} + lg(x))}{x^{4}ln^{3}{10}} + \frac{sin(x^{2} + lg(x))}{x^{4}ln^{4}{10}}\\ \end{split}\end{equation} \]

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