Derivative function：
Enter an original function (that is, the function to be derived), then set the variable to be derived and the order of the derivative, and click the "Next" button to obtain the derivative function of the corresponding order of the function.
Note that the input function supports mathematical functions and other constants.
Enter an original function (that is, the function to be derived), then set the variable to be derived and the order of the derivative, and click the "Next" button to obtain the derivative function of the corresponding order of the function.
Note that the input function supports mathematical functions and other constants.
Current location：Derivative function > Derivative function calculation history > Answer
There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {({198}^{{-168}^{cos(x)}})}^{\frac{1}{2}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = {198}^{(\frac{1}{2} * {-168}^{cos(x)})}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {198}^{(\frac{1}{2} * {-168}^{cos(x)})}\right)}{dx}\\=&({198}^{(\frac{1}{2} * {-168}^{cos(x)})}((\frac{1}{2}({-168}^{cos(x)}((-sin(x))ln(-168) + \frac{(cos(x))(0)}{(-168)})))ln(198) + \frac{(\frac{1}{2} * {-168}^{cos(x)})(0)}{(198)}))\\=&\frac{-{-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln(198)ln(-168)sin(x)}{2}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-{-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln(198)ln(-168)sin(x)}{2}\right)}{dx}\\=&\frac{-({-168}^{cos(x)}((-sin(x))ln(-168) + \frac{(cos(x))(0)}{(-168)})){198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln(198)ln(-168)sin(x)}{2} - \frac{{-168}^{cos(x)}({198}^{(\frac{1}{2} * {-168}^{cos(x)})}((\frac{1}{2}({-168}^{cos(x)}((-sin(x))ln(-168) + \frac{(cos(x))(0)}{(-168)})))ln(198) + \frac{(\frac{1}{2} * {-168}^{cos(x)})(0)}{(198)}))ln(198)ln(-168)sin(x)}{2} - \frac{{-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}*0ln(-168)sin(x)}{2(198)} - \frac{{-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln(198)*0sin(x)}{2(-168)} - \frac{{-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln(198)ln(-168)cos(x)}{2}\\=&\frac{{-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln(198)sin^{2}(x)}{2} + \frac{{-168}^{(2(cos(x)))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln^{2}(198)sin^{2}(x)}{4} - \frac{{198}^{(\frac{1}{2} * {-168}^{cos(x)})}{-168}^{cos(x)}ln(-168)ln(198)cos(x)}{2}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{{-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln(198)sin^{2}(x)}{2} + \frac{{-168}^{(2(cos(x)))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln^{2}(198)sin^{2}(x)}{4} - \frac{{198}^{(\frac{1}{2} * {-168}^{cos(x)})}{-168}^{cos(x)}ln(-168)ln(198)cos(x)}{2}\right)}{dx}\\=&\frac{({-168}^{cos(x)}((-sin(x))ln(-168) + \frac{(cos(x))(0)}{(-168)})){198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln(198)sin^{2}(x)}{2} + \frac{{-168}^{cos(x)}({198}^{(\frac{1}{2} * {-168}^{cos(x)})}((\frac{1}{2}({-168}^{cos(x)}((-sin(x))ln(-168) + \frac{(cos(x))(0)}{(-168)})))ln(198) + \frac{(\frac{1}{2} * {-168}^{cos(x)})(0)}{(198)}))ln^{2}(-168)ln(198)sin^{2}(x)}{2} + \frac{{-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}*2ln(-168)*0ln(198)sin^{2}(x)}{2(-168)} + \frac{{-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)*0sin^{2}(x)}{2(198)} + \frac{{-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln(198)*2sin(x)cos(x)}{2} + \frac{({-168}^{(2(cos(x)))}((2(-sin(x)))ln(-168) + \frac{(2(cos(x)))(0)}{(-168)})){198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln^{2}(198)sin^{2}(x)}{4} + \frac{{-168}^{(2(cos(x)))}({198}^{(\frac{1}{2} * {-168}^{cos(x)})}((\frac{1}{2}({-168}^{cos(x)}((-sin(x))ln(-168) + \frac{(cos(x))(0)}{(-168)})))ln(198) + \frac{(\frac{1}{2} * {-168}^{cos(x)})(0)}{(198)}))ln^{2}(-168)ln^{2}(198)sin^{2}(x)}{4} + \frac{{-168}^{(2(cos(x)))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}*2ln(-168)*0ln^{2}(198)sin^{2}(x)}{4(-168)} + \frac{{-168}^{(2(cos(x)))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)*2ln(198)*0sin^{2}(x)}{4(198)} + \frac{{-168}^{(2(cos(x)))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln^{2}(198)*2sin(x)cos(x)}{4} - \frac{({198}^{(\frac{1}{2} * {-168}^{cos(x)})}((\frac{1}{2}({-168}^{cos(x)}((-sin(x))ln(-168) + \frac{(cos(x))(0)}{(-168)})))ln(198) + \frac{(\frac{1}{2} * {-168}^{cos(x)})(0)}{(198)})){-168}^{cos(x)}ln(-168)ln(198)cos(x)}{2} - \frac{{198}^{(\frac{1}{2} * {-168}^{cos(x)})}({-168}^{cos(x)}((-sin(x))ln(-168) + \frac{(cos(x))(0)}{(-168)}))ln(-168)ln(198)cos(x)}{2} - \frac{{198}^{(\frac{1}{2} * {-168}^{cos(x)})}{-168}^{cos(x)}*0ln(198)cos(x)}{2(-168)} - \frac{{198}^{(\frac{1}{2} * {-168}^{cos(x)})}{-168}^{cos(x)}ln(-168)*0cos(x)}{2(198)} - \frac{{198}^{(\frac{1}{2} * {-168}^{cos(x)})}{-168}^{cos(x)}ln(-168)ln(198)*-sin(x)}{2}\\=&\frac{3 * {-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln(198)sin(x)cos(x)}{2} + \frac{3 * {-168}^{(2cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln^{2}(198)sin(x)cos(x)}{4} - \frac{3 * {-168}^{(2cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{3}(-168)ln^{2}(198)sin^{3}(x)}{4} - \frac{{-168}^{(3cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{3}(-168)ln^{3}(198)sin^{3}(x)}{8} - \frac{{-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{3}(-168)ln(198)sin^{3}(x)}{2} + \frac{{198}^{(\frac{1}{2} * {-168}^{cos(x)})}{-168}^{cos(x)}ln(-168)ln(198)sin(x)}{2}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3 * {-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln(198)sin(x)cos(x)}{2} + \frac{3 * {-168}^{(2cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln^{2}(198)sin(x)cos(x)}{4} - \frac{3 * {-168}^{(2cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{3}(-168)ln^{2}(198)sin^{3}(x)}{4} - \frac{{-168}^{(3cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{3}(-168)ln^{3}(198)sin^{3}(x)}{8} - \frac{{-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{3}(-168)ln(198)sin^{3}(x)}{2} + \frac{{198}^{(\frac{1}{2} * {-168}^{cos(x)})}{-168}^{cos(x)}ln(-168)ln(198)sin(x)}{2}\right)}{dx}\\=&\frac{3({-168}^{cos(x)}((-sin(x))ln(-168) + \frac{(cos(x))(0)}{(-168)})){198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln(198)sin(x)cos(x)}{2} + \frac{3 * {-168}^{cos(x)}({198}^{(\frac{1}{2} * {-168}^{cos(x)})}((\frac{1}{2}({-168}^{cos(x)}((-sin(x))ln(-168) + \frac{(cos(x))(0)}{(-168)})))ln(198) + \frac{(\frac{1}{2} * {-168}^{cos(x)})(0)}{(198)}))ln^{2}(-168)ln(198)sin(x)cos(x)}{2} + \frac{3 * {-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}*2ln(-168)*0ln(198)sin(x)cos(x)}{2(-168)} + \frac{3 * {-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)*0sin(x)cos(x)}{2(198)} + \frac{3 * {-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln(198)cos(x)cos(x)}{2} + \frac{3 * {-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln(198)sin(x)*-sin(x)}{2} + \frac{3({-168}^{(2cos(x))}((2*-sin(x))ln(-168) + \frac{(2cos(x))(0)}{(-168)})){198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln^{2}(198)sin(x)cos(x)}{4} + \frac{3 * {-168}^{(2cos(x))}({198}^{(\frac{1}{2} * {-168}^{cos(x)})}((\frac{1}{2}({-168}^{cos(x)}((-sin(x))ln(-168) + \frac{(cos(x))(0)}{(-168)})))ln(198) + \frac{(\frac{1}{2} * {-168}^{cos(x)})(0)}{(198)}))ln^{2}(-168)ln^{2}(198)sin(x)cos(x)}{4} + \frac{3 * {-168}^{(2cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}*2ln(-168)*0ln^{2}(198)sin(x)cos(x)}{4(-168)} + \frac{3 * {-168}^{(2cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)*2ln(198)*0sin(x)cos(x)}{4(198)} + \frac{3 * {-168}^{(2cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln^{2}(198)cos(x)cos(x)}{4} + \frac{3 * {-168}^{(2cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln^{2}(198)sin(x)*-sin(x)}{4} - \frac{3({-168}^{(2cos(x))}((2*-sin(x))ln(-168) + \frac{(2cos(x))(0)}{(-168)})){198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{3}(-168)ln^{2}(198)sin^{3}(x)}{4} - \frac{3 * {-168}^{(2cos(x))}({198}^{(\frac{1}{2} * {-168}^{cos(x)})}((\frac{1}{2}({-168}^{cos(x)}((-sin(x))ln(-168) + \frac{(cos(x))(0)}{(-168)})))ln(198) + \frac{(\frac{1}{2} * {-168}^{cos(x)})(0)}{(198)}))ln^{3}(-168)ln^{2}(198)sin^{3}(x)}{4} - \frac{3 * {-168}^{(2cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}*3ln^{2}(-168)*0ln^{2}(198)sin^{3}(x)}{4(-168)} - \frac{3 * {-168}^{(2cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{3}(-168)*2ln(198)*0sin^{3}(x)}{4(198)} - \frac{3 * {-168}^{(2cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{3}(-168)ln^{2}(198)*3sin^{2}(x)cos(x)}{4} - \frac{({-168}^{(3cos(x))}((3*-sin(x))ln(-168) + \frac{(3cos(x))(0)}{(-168)})){198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{3}(-168)ln^{3}(198)sin^{3}(x)}{8} - \frac{{-168}^{(3cos(x))}({198}^{(\frac{1}{2} * {-168}^{cos(x)})}((\frac{1}{2}({-168}^{cos(x)}((-sin(x))ln(-168) + \frac{(cos(x))(0)}{(-168)})))ln(198) + \frac{(\frac{1}{2} * {-168}^{cos(x)})(0)}{(198)}))ln^{3}(-168)ln^{3}(198)sin^{3}(x)}{8} - \frac{{-168}^{(3cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}*3ln^{2}(-168)*0ln^{3}(198)sin^{3}(x)}{8(-168)} - \frac{{-168}^{(3cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{3}(-168)*3ln^{2}(198)*0sin^{3}(x)}{8(198)} - \frac{{-168}^{(3cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{3}(-168)ln^{3}(198)*3sin^{2}(x)cos(x)}{8} - \frac{({-168}^{cos(x)}((-sin(x))ln(-168) + \frac{(cos(x))(0)}{(-168)})){198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{3}(-168)ln(198)sin^{3}(x)}{2} - \frac{{-168}^{cos(x)}({198}^{(\frac{1}{2} * {-168}^{cos(x)})}((\frac{1}{2}({-168}^{cos(x)}((-sin(x))ln(-168) + \frac{(cos(x))(0)}{(-168)})))ln(198) + \frac{(\frac{1}{2} * {-168}^{cos(x)})(0)}{(198)}))ln^{3}(-168)ln(198)sin^{3}(x)}{2} - \frac{{-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}*3ln^{2}(-168)*0ln(198)sin^{3}(x)}{2(-168)} - \frac{{-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{3}(-168)*0sin^{3}(x)}{2(198)} - \frac{{-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{3}(-168)ln(198)*3sin^{2}(x)cos(x)}{2} + \frac{({198}^{(\frac{1}{2} * {-168}^{cos(x)})}((\frac{1}{2}({-168}^{cos(x)}((-sin(x))ln(-168) + \frac{(cos(x))(0)}{(-168)})))ln(198) + \frac{(\frac{1}{2} * {-168}^{cos(x)})(0)}{(198)})){-168}^{cos(x)}ln(-168)ln(198)sin(x)}{2} + \frac{{198}^{(\frac{1}{2} * {-168}^{cos(x)})}({-168}^{cos(x)}((-sin(x))ln(-168) + \frac{(cos(x))(0)}{(-168)}))ln(-168)ln(198)sin(x)}{2} + \frac{{198}^{(\frac{1}{2} * {-168}^{cos(x)})}{-168}^{cos(x)}*0ln(198)sin(x)}{2(-168)} + \frac{{198}^{(\frac{1}{2} * {-168}^{cos(x)})}{-168}^{cos(x)}ln(-168)*0sin(x)}{2(198)} + \frac{{198}^{(\frac{1}{2} * {-168}^{cos(x)})}{-168}^{cos(x)}ln(-168)ln(198)cos(x)}{2}\\=&-3 * {-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{3}(-168)ln(198)sin^{2}(x)cos(x) - \frac{3 * {-168}^{(2(cos(x)))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{3}(-168)ln^{2}(198)sin^{2}(x)cos(x)}{4} + \frac{3 * {198}^{(\frac{1}{2} * {-168}^{cos(x)})}{-168}^{cos(x)}ln(198)ln^{2}(-168)cos^{2}(x)}{2} - \frac{15 * {-168}^{(2cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{3}(-168)ln^{2}(198)sin^{2}(x)cos(x)}{4} - \frac{3 * {-168}^{(3cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{3}(-168)ln^{3}(198)sin^{2}(x)cos(x)}{4} + \frac{3 * {198}^{(\frac{1}{2} * {-168}^{cos(x)})}{-168}^{(2cos(x))}ln^{2}(198)ln^{2}(-168)cos^{2}(x)}{4} + \frac{3 * {-168}^{(2cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{4}(-168)ln^{2}(198)sin^{4}(x)}{2} + \frac{3 * {-168}^{(3cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{4}(-168)ln^{3}(198)sin^{4}(x)}{4} + \frac{{-168}^{(4cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{4}(-168)ln^{4}(198)sin^{4}(x)}{16} - \frac{3 * {-168}^{(2cos(x))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln^{2}(198)sin^{2}(x)}{4} + \frac{{-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{4}(-168)ln(198)sin^{4}(x)}{2} + \frac{{-168}^{(2(cos(x)))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{4}(-168)ln^{2}(198)sin^{4}(x)}{4} - 2 * {-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln(198)sin^{2}(x) - \frac{{-168}^{(2(cos(x)))}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln^{2}(-168)ln^{2}(198)sin^{2}(x)}{4} + \frac{{-168}^{cos(x)}{198}^{(\frac{1}{2} * {-168}^{cos(x)})}ln(198)ln(-168)cos(x)}{2}\\ \end{split}\end{equation} \]