There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {10}^{lg(sin(x))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {10}^{lg(sin(x))}\right)}{dx}\\=&({10}^{lg(sin(x))}((\frac{cos(x)}{ln{10}(sin(x))})ln(10) + \frac{(lg(sin(x)))(0)}{(10)}))\\=&\frac{{10}^{lg(sin(x))}ln(10)cos(x)}{ln{10}sin(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{{10}^{lg(sin(x))}ln(10)cos(x)}{ln{10}sin(x)}\right)}{dx}\\=&\frac{({10}^{lg(sin(x))}((\frac{cos(x)}{ln{10}(sin(x))})ln(10) + \frac{(lg(sin(x)))(0)}{(10)}))ln(10)cos(x)}{ln{10}sin(x)} + \frac{{10}^{lg(sin(x))}*0cos(x)}{(10)ln{10}sin(x)} + \frac{{10}^{lg(sin(x))}ln(10)*-0cos(x)}{ln^{2}{10}sin(x)} + \frac{{10}^{lg(sin(x))}ln(10)*-cos(x)cos(x)}{ln{10}sin^{2}(x)} + \frac{{10}^{lg(sin(x))}ln(10)*-sin(x)}{ln{10}sin(x)}\\=&\frac{{10}^{lg(sin(x))}ln^{2}(10)cos^{2}(x)}{ln^{2}{10}sin^{2}(x)} - \frac{{10}^{lg(sin(x))}ln(10)cos^{2}(x)}{ln{10}sin^{2}(x)} - \frac{{10}^{lg(sin(x))}ln(10)}{ln{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{{10}^{lg(sin(x))}ln^{2}(10)cos^{2}(x)}{ln^{2}{10}sin^{2}(x)} - \frac{{10}^{lg(sin(x))}ln(10)cos^{2}(x)}{ln{10}sin^{2}(x)} - \frac{{10}^{lg(sin(x))}ln(10)}{ln{10}}\right)}{dx}\\=&\frac{({10}^{lg(sin(x))}((\frac{cos(x)}{ln{10}(sin(x))})ln(10) + \frac{(lg(sin(x)))(0)}{(10)}))ln^{2}(10)cos^{2}(x)}{ln^{2}{10}sin^{2}(x)} + \frac{{10}^{lg(sin(x))}*2ln(10)*0cos^{2}(x)}{(10)ln^{2}{10}sin^{2}(x)} + \frac{{10}^{lg(sin(x))}ln^{2}(10)*-2*0cos^{2}(x)}{ln^{3}{10}sin^{2}(x)} + \frac{{10}^{lg(sin(x))}ln^{2}(10)*-2cos(x)cos^{2}(x)}{ln^{2}{10}sin^{3}(x)} + \frac{{10}^{lg(sin(x))}ln^{2}(10)*-2cos(x)sin(x)}{ln^{2}{10}sin^{2}(x)} - \frac{({10}^{lg(sin(x))}((\frac{cos(x)}{ln{10}(sin(x))})ln(10) + \frac{(lg(sin(x)))(0)}{(10)}))ln(10)cos^{2}(x)}{ln{10}sin^{2}(x)} - \frac{{10}^{lg(sin(x))}*0cos^{2}(x)}{(10)ln{10}sin^{2}(x)} - \frac{{10}^{lg(sin(x))}ln(10)*-0cos^{2}(x)}{ln^{2}{10}sin^{2}(x)} - \frac{{10}^{lg(sin(x))}ln(10)*-2cos(x)cos^{2}(x)}{ln{10}sin^{3}(x)} - \frac{{10}^{lg(sin(x))}ln(10)*-2cos(x)sin(x)}{ln{10}sin^{2}(x)} - \frac{({10}^{lg(sin(x))}((\frac{cos(x)}{ln{10}(sin(x))})ln(10) + \frac{(lg(sin(x)))(0)}{(10)}))ln(10)}{ln{10}} - \frac{{10}^{lg(sin(x))}*0}{(10)ln{10}} - \frac{{10}^{lg(sin(x))}ln(10)*-0}{ln^{2}{10}}\\=&\frac{{10}^{lg(sin(x))}ln^{3}(10)cos^{3}(x)}{ln^{3}{10}sin^{3}(x)} - \frac{3 * {10}^{lg(sin(x))}ln^{2}(10)cos^{3}(x)}{ln^{2}{10}sin^{3}(x)} + \frac{2 * {10}^{lg(sin(x))}ln(10)cos^{3}(x)}{ln{10}sin^{3}(x)} - \frac{3 * {10}^{lg(sin(x))}ln^{2}(10)cos(x)}{ln^{2}{10}sin(x)} + \frac{2 * {10}^{lg(sin(x))}ln(10)cos(x)}{ln{10}sin(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{{10}^{lg(sin(x))}ln^{3}(10)cos^{3}(x)}{ln^{3}{10}sin^{3}(x)} - \frac{3 * {10}^{lg(sin(x))}ln^{2}(10)cos^{3}(x)}{ln^{2}{10}sin^{3}(x)} + \frac{2 * {10}^{lg(sin(x))}ln(10)cos^{3}(x)}{ln{10}sin^{3}(x)} - \frac{3 * {10}^{lg(sin(x))}ln^{2}(10)cos(x)}{ln^{2}{10}sin(x)} + \frac{2 * {10}^{lg(sin(x))}ln(10)cos(x)}{ln{10}sin(x)}\right)}{dx}\\=&\frac{({10}^{lg(sin(x))}((\frac{cos(x)}{ln{10}(sin(x))})ln(10) + \frac{(lg(sin(x)))(0)}{(10)}))ln^{3}(10)cos^{3}(x)}{ln^{3}{10}sin^{3}(x)} + \frac{{10}^{lg(sin(x))}*3ln^{2}(10)*0cos^{3}(x)}{(10)ln^{3}{10}sin^{3}(x)} + \frac{{10}^{lg(sin(x))}ln^{3}(10)*-3*0cos^{3}(x)}{ln^{4}{10}sin^{3}(x)} + \frac{{10}^{lg(sin(x))}ln^{3}(10)*-3cos(x)cos^{3}(x)}{ln^{3}{10}sin^{4}(x)} + \frac{{10}^{lg(sin(x))}ln^{3}(10)*-3cos^{2}(x)sin(x)}{ln^{3}{10}sin^{3}(x)} - \frac{3({10}^{lg(sin(x))}((\frac{cos(x)}{ln{10}(sin(x))})ln(10) + \frac{(lg(sin(x)))(0)}{(10)}))ln^{2}(10)cos^{3}(x)}{ln^{2}{10}sin^{3}(x)} - \frac{3 * {10}^{lg(sin(x))}*2ln(10)*0cos^{3}(x)}{(10)ln^{2}{10}sin^{3}(x)} - \frac{3 * {10}^{lg(sin(x))}ln^{2}(10)*-2*0cos^{3}(x)}{ln^{3}{10}sin^{3}(x)} - \frac{3 * {10}^{lg(sin(x))}ln^{2}(10)*-3cos(x)cos^{3}(x)}{ln^{2}{10}sin^{4}(x)} - \frac{3 * {10}^{lg(sin(x))}ln^{2}(10)*-3cos^{2}(x)sin(x)}{ln^{2}{10}sin^{3}(x)} + \frac{2({10}^{lg(sin(x))}((\frac{cos(x)}{ln{10}(sin(x))})ln(10) + \frac{(lg(sin(x)))(0)}{(10)}))ln(10)cos^{3}(x)}{ln{10}sin^{3}(x)} + \frac{2 * {10}^{lg(sin(x))}*0cos^{3}(x)}{(10)ln{10}sin^{3}(x)} + \frac{2 * {10}^{lg(sin(x))}ln(10)*-0cos^{3}(x)}{ln^{2}{10}sin^{3}(x)} + \frac{2 * {10}^{lg(sin(x))}ln(10)*-3cos(x)cos^{3}(x)}{ln{10}sin^{4}(x)} + \frac{2 * {10}^{lg(sin(x))}ln(10)*-3cos^{2}(x)sin(x)}{ln{10}sin^{3}(x)} - \frac{3({10}^{lg(sin(x))}((\frac{cos(x)}{ln{10}(sin(x))})ln(10) + \frac{(lg(sin(x)))(0)}{(10)}))ln^{2}(10)cos(x)}{ln^{2}{10}sin(x)} - \frac{3 * {10}^{lg(sin(x))}*2ln(10)*0cos(x)}{(10)ln^{2}{10}sin(x)} - \frac{3 * {10}^{lg(sin(x))}ln^{2}(10)*-2*0cos(x)}{ln^{3}{10}sin(x)} - \frac{3 * {10}^{lg(sin(x))}ln^{2}(10)*-cos(x)cos(x)}{ln^{2}{10}sin^{2}(x)} - \frac{3 * {10}^{lg(sin(x))}ln^{2}(10)*-sin(x)}{ln^{2}{10}sin(x)} + \frac{2({10}^{lg(sin(x))}((\frac{cos(x)}{ln{10}(sin(x))})ln(10) + \frac{(lg(sin(x)))(0)}{(10)}))ln(10)cos(x)}{ln{10}sin(x)} + \frac{2 * {10}^{lg(sin(x))}*0cos(x)}{(10)ln{10}sin(x)} + \frac{2 * {10}^{lg(sin(x))}ln(10)*-0cos(x)}{ln^{2}{10}sin(x)} + \frac{2 * {10}^{lg(sin(x))}ln(10)*-cos(x)cos(x)}{ln{10}sin^{2}(x)} + \frac{2 * {10}^{lg(sin(x))}ln(10)*-sin(x)}{ln{10}sin(x)}\\=&\frac{{10}^{lg(sin(x))}ln^{4}(10)cos^{4}(x)}{ln^{4}{10}sin^{4}(x)} - \frac{6 * {10}^{lg(sin(x))}ln^{3}(10)cos^{4}(x)}{ln^{3}{10}sin^{4}(x)} + \frac{11 * {10}^{lg(sin(x))}ln^{2}(10)cos^{4}(x)}{ln^{2}{10}sin^{4}(x)} - \frac{6 * {10}^{lg(sin(x))}ln^{3}(10)cos^{2}(x)}{ln^{3}{10}sin^{2}(x)} - \frac{6 * {10}^{lg(sin(x))}ln(10)cos^{4}(x)}{ln{10}sin^{4}(x)} + \frac{14 * {10}^{lg(sin(x))}ln^{2}(10)cos^{2}(x)}{ln^{2}{10}sin^{2}(x)} - \frac{8 * {10}^{lg(sin(x))}ln(10)cos^{2}(x)}{ln{10}sin^{2}(x)} + \frac{3 * {10}^{lg(sin(x))}ln^{2}(10)}{ln^{2}{10}} - \frac{2 * {10}^{lg(sin(x))}ln(10)}{ln{10}}\\ \end{split}\end{equation} \]

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