There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {({193}^{(-({168}^{cos(x)}))})}^{\frac{1}{2}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = {193}^{(\frac{-1}{2} * {168}^{cos(x)})}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {193}^{(\frac{-1}{2} * {168}^{cos(x)})}\right)}{dx}\\=&({193}^{(\frac{-1}{2} * {168}^{cos(x)})}((\frac{-1}{2}({168}^{cos(x)}((-sin(x))ln(168) + \frac{(cos(x))(0)}{(168)})))ln(193) + \frac{(\frac{-1}{2} * {168}^{cos(x)})(0)}{(193)}))\\=&\frac{{168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln(193)ln(168)sin(x)}{2}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{{168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln(193)ln(168)sin(x)}{2}\right)}{dx}\\=&\frac{({168}^{cos(x)}((-sin(x))ln(168) + \frac{(cos(x))(0)}{(168)})){193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln(193)ln(168)sin(x)}{2} + \frac{{168}^{cos(x)}({193}^{(\frac{-1}{2} * {168}^{cos(x)})}((\frac{-1}{2}({168}^{cos(x)}((-sin(x))ln(168) + \frac{(cos(x))(0)}{(168)})))ln(193) + \frac{(\frac{-1}{2} * {168}^{cos(x)})(0)}{(193)}))ln(193)ln(168)sin(x)}{2} + \frac{{168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}*0ln(168)sin(x)}{2(193)} + \frac{{168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln(193)*0sin(x)}{2(168)} + \frac{{168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln(193)ln(168)cos(x)}{2}\\=&\frac{-{168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln(193)sin^{2}(x)}{2} + \frac{{168}^{(2(cos(x)))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln^{2}(193)sin^{2}(x)}{4} + \frac{{193}^{(\frac{-1}{2} * {168}^{cos(x)})}{168}^{cos(x)}ln(168)ln(193)cos(x)}{2}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-{168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln(193)sin^{2}(x)}{2} + \frac{{168}^{(2(cos(x)))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln^{2}(193)sin^{2}(x)}{4} + \frac{{193}^{(\frac{-1}{2} * {168}^{cos(x)})}{168}^{cos(x)}ln(168)ln(193)cos(x)}{2}\right)}{dx}\\=&\frac{-({168}^{cos(x)}((-sin(x))ln(168) + \frac{(cos(x))(0)}{(168)})){193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln(193)sin^{2}(x)}{2} - \frac{{168}^{cos(x)}({193}^{(\frac{-1}{2} * {168}^{cos(x)})}((\frac{-1}{2}({168}^{cos(x)}((-sin(x))ln(168) + \frac{(cos(x))(0)}{(168)})))ln(193) + \frac{(\frac{-1}{2} * {168}^{cos(x)})(0)}{(193)}))ln^{2}(168)ln(193)sin^{2}(x)}{2} - \frac{{168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}*2ln(168)*0ln(193)sin^{2}(x)}{2(168)} - \frac{{168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)*0sin^{2}(x)}{2(193)} - \frac{{168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln(193)*2sin(x)cos(x)}{2} + \frac{({168}^{(2(cos(x)))}((2(-sin(x)))ln(168) + \frac{(2(cos(x)))(0)}{(168)})){193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln^{2}(193)sin^{2}(x)}{4} + \frac{{168}^{(2(cos(x)))}({193}^{(\frac{-1}{2} * {168}^{cos(x)})}((\frac{-1}{2}({168}^{cos(x)}((-sin(x))ln(168) + \frac{(cos(x))(0)}{(168)})))ln(193) + \frac{(\frac{-1}{2} * {168}^{cos(x)})(0)}{(193)}))ln^{2}(168)ln^{2}(193)sin^{2}(x)}{4} + \frac{{168}^{(2(cos(x)))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}*2ln(168)*0ln^{2}(193)sin^{2}(x)}{4(168)} + \frac{{168}^{(2(cos(x)))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)*2ln(193)*0sin^{2}(x)}{4(193)} + \frac{{168}^{(2(cos(x)))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln^{2}(193)*2sin(x)cos(x)}{4} + \frac{({193}^{(\frac{-1}{2} * {168}^{cos(x)})}((\frac{-1}{2}({168}^{cos(x)}((-sin(x))ln(168) + \frac{(cos(x))(0)}{(168)})))ln(193) + \frac{(\frac{-1}{2} * {168}^{cos(x)})(0)}{(193)})){168}^{cos(x)}ln(168)ln(193)cos(x)}{2} + \frac{{193}^{(\frac{-1}{2} * {168}^{cos(x)})}({168}^{cos(x)}((-sin(x))ln(168) + \frac{(cos(x))(0)}{(168)}))ln(168)ln(193)cos(x)}{2} + \frac{{193}^{(\frac{-1}{2} * {168}^{cos(x)})}{168}^{cos(x)}*0ln(193)cos(x)}{2(168)} + \frac{{193}^{(\frac{-1}{2} * {168}^{cos(x)})}{168}^{cos(x)}ln(168)*0cos(x)}{2(193)} + \frac{{193}^{(\frac{-1}{2} * {168}^{cos(x)})}{168}^{cos(x)}ln(168)ln(193)*-sin(x)}{2}\\=& - \frac{3 * {168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln(193)sin(x)cos(x)}{2} + \frac{3 * {168}^{(2cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln^{2}(193)sin(x)cos(x)}{4} - \frac{3 * {168}^{(2cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{3}(168)ln^{2}(193)sin^{3}(x)}{4} + \frac{{168}^{(3cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{3}(168)ln^{3}(193)sin^{3}(x)}{8} + \frac{{168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{3}(168)ln(193)sin^{3}(x)}{2} - \frac{{193}^{(\frac{-1}{2} * {168}^{cos(x)})}{168}^{cos(x)}ln(168)ln(193)sin(x)}{2}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - \frac{3 * {168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln(193)sin(x)cos(x)}{2} + \frac{3 * {168}^{(2cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln^{2}(193)sin(x)cos(x)}{4} - \frac{3 * {168}^{(2cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{3}(168)ln^{2}(193)sin^{3}(x)}{4} + \frac{{168}^{(3cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{3}(168)ln^{3}(193)sin^{3}(x)}{8} + \frac{{168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{3}(168)ln(193)sin^{3}(x)}{2} - \frac{{193}^{(\frac{-1}{2} * {168}^{cos(x)})}{168}^{cos(x)}ln(168)ln(193)sin(x)}{2}\right)}{dx}\\=& - \frac{3({168}^{cos(x)}((-sin(x))ln(168) + \frac{(cos(x))(0)}{(168)})){193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln(193)sin(x)cos(x)}{2} - \frac{3 * {168}^{cos(x)}({193}^{(\frac{-1}{2} * {168}^{cos(x)})}((\frac{-1}{2}({168}^{cos(x)}((-sin(x))ln(168) + \frac{(cos(x))(0)}{(168)})))ln(193) + \frac{(\frac{-1}{2} * {168}^{cos(x)})(0)}{(193)}))ln^{2}(168)ln(193)sin(x)cos(x)}{2} - \frac{3 * {168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}*2ln(168)*0ln(193)sin(x)cos(x)}{2(168)} - \frac{3 * {168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)*0sin(x)cos(x)}{2(193)} - \frac{3 * {168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln(193)cos(x)cos(x)}{2} - \frac{3 * {168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln(193)sin(x)*-sin(x)}{2} + \frac{3({168}^{(2cos(x))}((2*-sin(x))ln(168) + \frac{(2cos(x))(0)}{(168)})){193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln^{2}(193)sin(x)cos(x)}{4} + \frac{3 * {168}^{(2cos(x))}({193}^{(\frac{-1}{2} * {168}^{cos(x)})}((\frac{-1}{2}({168}^{cos(x)}((-sin(x))ln(168) + \frac{(cos(x))(0)}{(168)})))ln(193) + \frac{(\frac{-1}{2} * {168}^{cos(x)})(0)}{(193)}))ln^{2}(168)ln^{2}(193)sin(x)cos(x)}{4} + \frac{3 * {168}^{(2cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}*2ln(168)*0ln^{2}(193)sin(x)cos(x)}{4(168)} + \frac{3 * {168}^{(2cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)*2ln(193)*0sin(x)cos(x)}{4(193)} + \frac{3 * {168}^{(2cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln^{2}(193)cos(x)cos(x)}{4} + \frac{3 * {168}^{(2cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln^{2}(193)sin(x)*-sin(x)}{4} - \frac{3({168}^{(2cos(x))}((2*-sin(x))ln(168) + \frac{(2cos(x))(0)}{(168)})){193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{3}(168)ln^{2}(193)sin^{3}(x)}{4} - \frac{3 * {168}^{(2cos(x))}({193}^{(\frac{-1}{2} * {168}^{cos(x)})}((\frac{-1}{2}({168}^{cos(x)}((-sin(x))ln(168) + \frac{(cos(x))(0)}{(168)})))ln(193) + \frac{(\frac{-1}{2} * {168}^{cos(x)})(0)}{(193)}))ln^{3}(168)ln^{2}(193)sin^{3}(x)}{4} - \frac{3 * {168}^{(2cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}*3ln^{2}(168)*0ln^{2}(193)sin^{3}(x)}{4(168)} - \frac{3 * {168}^{(2cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{3}(168)*2ln(193)*0sin^{3}(x)}{4(193)} - \frac{3 * {168}^{(2cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{3}(168)ln^{2}(193)*3sin^{2}(x)cos(x)}{4} + \frac{({168}^{(3cos(x))}((3*-sin(x))ln(168) + \frac{(3cos(x))(0)}{(168)})){193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{3}(168)ln^{3}(193)sin^{3}(x)}{8} + \frac{{168}^{(3cos(x))}({193}^{(\frac{-1}{2} * {168}^{cos(x)})}((\frac{-1}{2}({168}^{cos(x)}((-sin(x))ln(168) + \frac{(cos(x))(0)}{(168)})))ln(193) + \frac{(\frac{-1}{2} * {168}^{cos(x)})(0)}{(193)}))ln^{3}(168)ln^{3}(193)sin^{3}(x)}{8} + \frac{{168}^{(3cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}*3ln^{2}(168)*0ln^{3}(193)sin^{3}(x)}{8(168)} + \frac{{168}^{(3cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{3}(168)*3ln^{2}(193)*0sin^{3}(x)}{8(193)} + \frac{{168}^{(3cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{3}(168)ln^{3}(193)*3sin^{2}(x)cos(x)}{8} + \frac{({168}^{cos(x)}((-sin(x))ln(168) + \frac{(cos(x))(0)}{(168)})){193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{3}(168)ln(193)sin^{3}(x)}{2} + \frac{{168}^{cos(x)}({193}^{(\frac{-1}{2} * {168}^{cos(x)})}((\frac{-1}{2}({168}^{cos(x)}((-sin(x))ln(168) + \frac{(cos(x))(0)}{(168)})))ln(193) + \frac{(\frac{-1}{2} * {168}^{cos(x)})(0)}{(193)}))ln^{3}(168)ln(193)sin^{3}(x)}{2} + \frac{{168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}*3ln^{2}(168)*0ln(193)sin^{3}(x)}{2(168)} + \frac{{168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{3}(168)*0sin^{3}(x)}{2(193)} + \frac{{168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{3}(168)ln(193)*3sin^{2}(x)cos(x)}{2} - \frac{({193}^{(\frac{-1}{2} * {168}^{cos(x)})}((\frac{-1}{2}({168}^{cos(x)}((-sin(x))ln(168) + \frac{(cos(x))(0)}{(168)})))ln(193) + \frac{(\frac{-1}{2} * {168}^{cos(x)})(0)}{(193)})){168}^{cos(x)}ln(168)ln(193)sin(x)}{2} - \frac{{193}^{(\frac{-1}{2} * {168}^{cos(x)})}({168}^{cos(x)}((-sin(x))ln(168) + \frac{(cos(x))(0)}{(168)}))ln(168)ln(193)sin(x)}{2} - \frac{{193}^{(\frac{-1}{2} * {168}^{cos(x)})}{168}^{cos(x)}*0ln(193)sin(x)}{2(168)} - \frac{{193}^{(\frac{-1}{2} * {168}^{cos(x)})}{168}^{cos(x)}ln(168)*0sin(x)}{2(193)} - \frac{{193}^{(\frac{-1}{2} * {168}^{cos(x)})}{168}^{cos(x)}ln(168)ln(193)cos(x)}{2}\\=&3 * {168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{3}(168)ln(193)sin^{2}(x)cos(x) - \frac{3 * {168}^{(2(cos(x)))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{3}(168)ln^{2}(193)sin^{2}(x)cos(x)}{4} - \frac{3 * {193}^{(\frac{-1}{2} * {168}^{cos(x)})}{168}^{cos(x)}ln(193)ln^{2}(168)cos^{2}(x)}{2} - \frac{15 * {168}^{(2cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{3}(168)ln^{2}(193)sin^{2}(x)cos(x)}{4} + \frac{3 * {168}^{(3cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{3}(168)ln^{3}(193)sin^{2}(x)cos(x)}{4} + \frac{3 * {193}^{(\frac{-1}{2} * {168}^{cos(x)})}{168}^{(2cos(x))}ln^{2}(193)ln^{2}(168)cos^{2}(x)}{4} + \frac{3 * {168}^{(2cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{4}(168)ln^{2}(193)sin^{4}(x)}{2} - \frac{3 * {168}^{(3cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{4}(168)ln^{3}(193)sin^{4}(x)}{4} + \frac{{168}^{(4cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{4}(168)ln^{4}(193)sin^{4}(x)}{16} - \frac{3 * {168}^{(2cos(x))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln^{2}(193)sin^{2}(x)}{4} - \frac{{168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{4}(168)ln(193)sin^{4}(x)}{2} + \frac{{168}^{(2(cos(x)))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{4}(168)ln^{2}(193)sin^{4}(x)}{4} + 2 * {168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln(193)sin^{2}(x) - \frac{{168}^{(2(cos(x)))}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln^{2}(168)ln^{2}(193)sin^{2}(x)}{4} - \frac{{168}^{cos(x)}{193}^{(\frac{-1}{2} * {168}^{cos(x)})}ln(193)ln(168)cos(x)}{2}\\ \end{split}\end{equation} \]

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