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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{(x + x - xx)}{e^{lg(sin(x))}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{2x}{e^{lg(sin(x))}} - \frac{x^{2}}{e^{lg(sin(x))}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{2x}{e^{lg(sin(x))}} - \frac{x^{2}}{e^{lg(sin(x))}}\right)}{dx}\\=&\frac{2}{e^{lg(sin(x))}} + \frac{2x*-e^{lg(sin(x))}cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))} - \frac{2x}{e^{lg(sin(x))}} - \frac{x^{2}*-e^{lg(sin(x))}cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))}\\=&\frac{2}{e^{lg(sin(x))}} - \frac{2xcos(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{x^{2}cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{2x}{e^{lg(sin(x))}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{e^{lg(sin(x))}} - \frac{2xcos(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{x^{2}cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{2x}{e^{lg(sin(x))}}\right)}{dx}\\=&\frac{2*-e^{lg(sin(x))}cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))} - \frac{2cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{2x*-e^{lg(sin(x))}cos(x)cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin(x)} - \frac{2x*-0cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} - \frac{2x*-cos(x)cos(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} - \frac{2x*-sin(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{2xcos(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{x^{2}*-e^{lg(sin(x))}cos(x)cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin(x)} + \frac{x^{2}*-0cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} + \frac{x^{2}*-cos(x)cos(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{x^{2}*-sin(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{2}{e^{lg(sin(x))}} - \frac{2x*-e^{lg(sin(x))}cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))}\\=&\frac{-4cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{2xcos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{2xcos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{4xcos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{x^{2}cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} - \frac{x^{2}cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{2x}{e^{lg(sin(x))}ln{10}} - \frac{x^{2}}{e^{lg(sin(x))}ln{10}} - \frac{2}{e^{lg(sin(x))}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-4cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{2xcos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{2xcos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{4xcos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{x^{2}cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} - \frac{x^{2}cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{2x}{e^{lg(sin(x))}ln{10}} - \frac{x^{2}}{e^{lg(sin(x))}ln{10}} - \frac{2}{e^{lg(sin(x))}}\right)}{dx}\\=&\frac{-4*-e^{lg(sin(x))}cos(x)cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin(x)} - \frac{4*-0cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} - \frac{4*-cos(x)cos(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} - \frac{4*-sin(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{2cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{2x*-e^{lg(sin(x))}cos(x)cos^{2}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln^{2}{10}sin^{2}(x)} + \frac{2x*-2*0cos^{2}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{2}(x)} + \frac{2x*-2cos(x)cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} + \frac{2x*-2cos(x)sin(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{2cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{2x*-e^{lg(sin(x))}cos(x)cos^{2}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin^{2}(x)} + \frac{2x*-0cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{2x*-2cos(x)cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} + \frac{2x*-2cos(x)sin(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{4cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{4x*-e^{lg(sin(x))}cos(x)cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin(x)} + \frac{4x*-0cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} + \frac{4x*-cos(x)cos(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{4x*-sin(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{2xcos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} - \frac{x^{2}*-e^{lg(sin(x))}cos(x)cos^{2}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln^{2}{10}sin^{2}(x)} - \frac{x^{2}*-2*0cos^{2}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{2}(x)} - \frac{x^{2}*-2cos(x)cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{x^{2}*-2cos(x)sin(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} - \frac{2xcos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} - \frac{x^{2}*-e^{lg(sin(x))}cos(x)cos^{2}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin^{2}(x)} - \frac{x^{2}*-0cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} - \frac{x^{2}*-2cos(x)cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} - \frac{x^{2}*-2cos(x)sin(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{2}{e^{lg(sin(x))}ln{10}} + \frac{2x*-e^{lg(sin(x))}cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}} + \frac{2x*-0}{e^{lg(sin(x))}ln^{2}{10}} - \frac{2x}{e^{lg(sin(x))}ln{10}} - \frac{x^{2}*-e^{lg(sin(x))}cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}} - \frac{x^{2}*-0}{e^{lg(sin(x))}ln^{2}{10}} - \frac{2*-e^{lg(sin(x))}cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))}\\=&\frac{6cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{6cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{6cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{2xcos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} - \frac{6xcos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{4xcos^{3}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} - \frac{6xcos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} - \frac{6xcos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{3x^{2}cos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{4xcos(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{6}{e^{lg(sin(x))}ln{10}} - \frac{6xcos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{2x^{2}cos^{3}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} + \frac{3x^{2}cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} + \frac{x^{2}cos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} + \frac{2x^{2}cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{6x}{e^{lg(sin(x))}ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{6cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{6cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{6cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{2xcos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} - \frac{6xcos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{4xcos^{3}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} - \frac{6xcos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} - \frac{6xcos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{3x^{2}cos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{4xcos(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{6}{e^{lg(sin(x))}ln{10}} - \frac{6xcos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{2x^{2}cos^{3}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} + \frac{3x^{2}cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} + \frac{x^{2}cos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} + \frac{2x^{2}cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{6x}{e^{lg(sin(x))}ln{10}}\right)}{dx}\\=&\frac{6*-e^{lg(sin(x))}cos(x)cos^{2}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln^{2}{10}sin^{2}(x)} + \frac{6*-2*0cos^{2}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{2}(x)} + \frac{6*-2cos(x)cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} + \frac{6*-2cos(x)sin(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{6*-e^{lg(sin(x))}cos(x)cos^{2}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin^{2}(x)} + \frac{6*-0cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{6*-2cos(x)cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} + \frac{6*-2cos(x)sin(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{6*-e^{lg(sin(x))}cos(x)cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin(x)} + \frac{6*-0cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} + \frac{6*-cos(x)cos(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{6*-sin(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{2cos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} - \frac{2x*-e^{lg(sin(x))}cos(x)cos^{3}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln^{3}{10}sin^{3}(x)} - \frac{2x*-3*0cos^{3}(x)}{e^{lg(sin(x))}ln^{4}{10}sin^{3}(x)} - \frac{2x*-3cos(x)cos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{4}(x)} - \frac{2x*-3cos^{2}(x)sin(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} - \frac{6cos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{6x*-e^{lg(sin(x))}cos(x)cos^{3}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln^{2}{10}sin^{3}(x)} - \frac{6x*-2*0cos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} - \frac{6x*-3cos(x)cos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{4}(x)} - \frac{6x*-3cos^{2}(x)sin(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{4cos^{3}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} - \frac{4x*-e^{lg(sin(x))}cos(x)cos^{3}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin^{3}(x)} - \frac{4x*-0cos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{4x*-3cos(x)cos^{3}(x)}{e^{lg(sin(x))}ln{10}sin^{4}(x)} - \frac{4x*-3cos^{2}(x)sin(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} - \frac{6cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} - \frac{6x*-e^{lg(sin(x))}cos(x)cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln^{2}{10}sin(x)} - \frac{6x*-2*0cos(x)}{e^{lg(sin(x))}ln^{3}{10}sin(x)} - \frac{6x*-cos(x)cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} - \frac{6x*-sin(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} - \frac{6cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} - \frac{6x*-e^{lg(sin(x))}cos(x)cos^{2}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin^{2}(x)} - \frac{6x*-0cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} - \frac{6x*-2cos(x)cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} - \frac{6x*-2cos(x)sin(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{3*2xcos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} + \frac{3x^{2}*-e^{lg(sin(x))}cos(x)cos^{3}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln^{2}{10}sin^{3}(x)} + \frac{3x^{2}*-2*0cos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} + \frac{3x^{2}*-3cos(x)cos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{4}(x)} + \frac{3x^{2}*-3cos^{2}(x)sin(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{4cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{4x*-e^{lg(sin(x))}cos(x)cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin(x)} - \frac{4x*-0cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} - \frac{4x*-cos(x)cos(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} - \frac{4x*-sin(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{6*-e^{lg(sin(x))}cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}} + \frac{6*-0}{e^{lg(sin(x))}ln^{2}{10}} - \frac{6cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} - \frac{6x*-e^{lg(sin(x))}cos(x)cos^{2}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln^{2}{10}sin^{2}(x)} - \frac{6x*-2*0cos^{2}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{2}(x)} - \frac{6x*-2cos(x)cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{6x*-2cos(x)sin(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{2*2xcos^{3}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} + \frac{2x^{2}*-e^{lg(sin(x))}cos(x)cos^{3}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin^{3}(x)} + \frac{2x^{2}*-0cos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} + \frac{2x^{2}*-3cos(x)cos^{3}(x)}{e^{lg(sin(x))}ln{10}sin^{4}(x)} + \frac{2x^{2}*-3cos^{2}(x)sin(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} + \frac{3*2xcos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} + \frac{3x^{2}*-e^{lg(sin(x))}cos(x)cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln^{2}{10}sin(x)} + \frac{3x^{2}*-2*0cos(x)}{e^{lg(sin(x))}ln^{3}{10}sin(x)} + \frac{3x^{2}*-cos(x)cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{3x^{2}*-sin(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} + \frac{2xcos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} + \frac{x^{2}*-e^{lg(sin(x))}cos(x)cos^{3}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln^{3}{10}sin^{3}(x)} + \frac{x^{2}*-3*0cos^{3}(x)}{e^{lg(sin(x))}ln^{4}{10}sin^{3}(x)} + \frac{x^{2}*-3cos(x)cos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{4}(x)} + \frac{x^{2}*-3cos^{2}(x)sin(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} + \frac{2*2xcos(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{2x^{2}*-e^{lg(sin(x))}cos(x)cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin(x)} + \frac{2x^{2}*-0cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} + \frac{2x^{2}*-cos(x)cos(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{2x^{2}*-sin(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{6}{e^{lg(sin(x))}ln{10}} - \frac{6x*-e^{lg(sin(x))}cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}} - \frac{6x*-0}{e^{lg(sin(x))}ln^{2}{10}}\\=&\frac{-8cos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} - \frac{24cos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{24cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} - \frac{16cos^{3}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} - \frac{12cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} - \frac{16cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{12cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} - \frac{12}{e^{lg(sin(x))}ln{10}} + \frac{2xcos^{4}(x)}{e^{lg(sin(x))}ln^{4}{10}sin^{4}(x)} + \frac{12xcos^{4}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{4}(x)} + \frac{22xcos^{4}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{4}(x)} + \frac{12xcos^{2}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{2}(x)} + \frac{12xcos^{4}(x)}{e^{lg(sin(x))}ln{10}sin^{4}(x)} + \frac{28xcos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{16xcos^{3}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} - \frac{11x^{2}cos^{4}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{4}(x)} + \frac{16xcos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{24xcos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} + \frac{16xcos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{6x^{2}cos^{4}(x)}{e^{lg(sin(x))}ln{10}sin^{4}(x)} - \frac{14x^{2}cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} - \frac{6x^{2}cos^{4}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{4}(x)} - \frac{8x^{2}cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{24xcos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} + \frac{8xcos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} - \frac{6x^{2}cos^{2}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{2}(x)} - \frac{x^{2}cos^{4}(x)}{e^{lg(sin(x))}ln^{4}{10}sin^{4}(x)} - \frac{3x^{2}}{e^{lg(sin(x))}ln^{2}{10}} + \frac{4x}{e^{lg(sin(x))}ln{10}} + \frac{6x}{e^{lg(sin(x))}ln^{2}{10}} - \frac{2x^{2}}{e^{lg(sin(x))}ln{10}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{(x + x - xx)}{e^{lg(sin(x))}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{2x}{e^{lg(sin(x))}} - \frac{x^{2}}{e^{lg(sin(x))}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{2x}{e^{lg(sin(x))}} - \frac{x^{2}}{e^{lg(sin(x))}}\right)}{dx}\\=&\frac{2}{e^{lg(sin(x))}} + \frac{2x*-e^{lg(sin(x))}cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))} - \frac{2x}{e^{lg(sin(x))}} - \frac{x^{2}*-e^{lg(sin(x))}cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))}\\=&\frac{2}{e^{lg(sin(x))}} - \frac{2xcos(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{x^{2}cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{2x}{e^{lg(sin(x))}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{e^{lg(sin(x))}} - \frac{2xcos(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{x^{2}cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{2x}{e^{lg(sin(x))}}\right)}{dx}\\=&\frac{2*-e^{lg(sin(x))}cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))} - \frac{2cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{2x*-e^{lg(sin(x))}cos(x)cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin(x)} - \frac{2x*-0cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} - \frac{2x*-cos(x)cos(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} - \frac{2x*-sin(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{2xcos(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{x^{2}*-e^{lg(sin(x))}cos(x)cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin(x)} + \frac{x^{2}*-0cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} + \frac{x^{2}*-cos(x)cos(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{x^{2}*-sin(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{2}{e^{lg(sin(x))}} - \frac{2x*-e^{lg(sin(x))}cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))}\\=&\frac{-4cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{2xcos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{2xcos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{4xcos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{x^{2}cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} - \frac{x^{2}cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{2x}{e^{lg(sin(x))}ln{10}} - \frac{x^{2}}{e^{lg(sin(x))}ln{10}} - \frac{2}{e^{lg(sin(x))}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-4cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{2xcos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{2xcos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{4xcos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{x^{2}cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} - \frac{x^{2}cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{2x}{e^{lg(sin(x))}ln{10}} - \frac{x^{2}}{e^{lg(sin(x))}ln{10}} - \frac{2}{e^{lg(sin(x))}}\right)}{dx}\\=&\frac{-4*-e^{lg(sin(x))}cos(x)cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin(x)} - \frac{4*-0cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} - \frac{4*-cos(x)cos(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} - \frac{4*-sin(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{2cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{2x*-e^{lg(sin(x))}cos(x)cos^{2}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln^{2}{10}sin^{2}(x)} + \frac{2x*-2*0cos^{2}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{2}(x)} + \frac{2x*-2cos(x)cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} + \frac{2x*-2cos(x)sin(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{2cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{2x*-e^{lg(sin(x))}cos(x)cos^{2}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin^{2}(x)} + \frac{2x*-0cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{2x*-2cos(x)cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} + \frac{2x*-2cos(x)sin(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{4cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{4x*-e^{lg(sin(x))}cos(x)cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin(x)} + \frac{4x*-0cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} + \frac{4x*-cos(x)cos(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{4x*-sin(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{2xcos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} - \frac{x^{2}*-e^{lg(sin(x))}cos(x)cos^{2}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln^{2}{10}sin^{2}(x)} - \frac{x^{2}*-2*0cos^{2}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{2}(x)} - \frac{x^{2}*-2cos(x)cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{x^{2}*-2cos(x)sin(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} - \frac{2xcos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} - \frac{x^{2}*-e^{lg(sin(x))}cos(x)cos^{2}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin^{2}(x)} - \frac{x^{2}*-0cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} - \frac{x^{2}*-2cos(x)cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} - \frac{x^{2}*-2cos(x)sin(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{2}{e^{lg(sin(x))}ln{10}} + \frac{2x*-e^{lg(sin(x))}cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}} + \frac{2x*-0}{e^{lg(sin(x))}ln^{2}{10}} - \frac{2x}{e^{lg(sin(x))}ln{10}} - \frac{x^{2}*-e^{lg(sin(x))}cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}} - \frac{x^{2}*-0}{e^{lg(sin(x))}ln^{2}{10}} - \frac{2*-e^{lg(sin(x))}cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))}\\=&\frac{6cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{6cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{6cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{2xcos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} - \frac{6xcos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{4xcos^{3}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} - \frac{6xcos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} - \frac{6xcos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{3x^{2}cos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{4xcos(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{6}{e^{lg(sin(x))}ln{10}} - \frac{6xcos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{2x^{2}cos^{3}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} + \frac{3x^{2}cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} + \frac{x^{2}cos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} + \frac{2x^{2}cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{6x}{e^{lg(sin(x))}ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{6cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{6cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{6cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{2xcos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} - \frac{6xcos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{4xcos^{3}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} - \frac{6xcos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} - \frac{6xcos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{3x^{2}cos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{4xcos(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{6}{e^{lg(sin(x))}ln{10}} - \frac{6xcos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{2x^{2}cos^{3}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} + \frac{3x^{2}cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} + \frac{x^{2}cos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} + \frac{2x^{2}cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{6x}{e^{lg(sin(x))}ln{10}}\right)}{dx}\\=&\frac{6*-e^{lg(sin(x))}cos(x)cos^{2}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln^{2}{10}sin^{2}(x)} + \frac{6*-2*0cos^{2}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{2}(x)} + \frac{6*-2cos(x)cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} + \frac{6*-2cos(x)sin(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{6*-e^{lg(sin(x))}cos(x)cos^{2}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin^{2}(x)} + \frac{6*-0cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{6*-2cos(x)cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} + \frac{6*-2cos(x)sin(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{6*-e^{lg(sin(x))}cos(x)cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin(x)} + \frac{6*-0cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} + \frac{6*-cos(x)cos(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{6*-sin(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{2cos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} - \frac{2x*-e^{lg(sin(x))}cos(x)cos^{3}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln^{3}{10}sin^{3}(x)} - \frac{2x*-3*0cos^{3}(x)}{e^{lg(sin(x))}ln^{4}{10}sin^{3}(x)} - \frac{2x*-3cos(x)cos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{4}(x)} - \frac{2x*-3cos^{2}(x)sin(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} - \frac{6cos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{6x*-e^{lg(sin(x))}cos(x)cos^{3}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln^{2}{10}sin^{3}(x)} - \frac{6x*-2*0cos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} - \frac{6x*-3cos(x)cos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{4}(x)} - \frac{6x*-3cos^{2}(x)sin(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{4cos^{3}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} - \frac{4x*-e^{lg(sin(x))}cos(x)cos^{3}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin^{3}(x)} - \frac{4x*-0cos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{4x*-3cos(x)cos^{3}(x)}{e^{lg(sin(x))}ln{10}sin^{4}(x)} - \frac{4x*-3cos^{2}(x)sin(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} - \frac{6cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} - \frac{6x*-e^{lg(sin(x))}cos(x)cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln^{2}{10}sin(x)} - \frac{6x*-2*0cos(x)}{e^{lg(sin(x))}ln^{3}{10}sin(x)} - \frac{6x*-cos(x)cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} - \frac{6x*-sin(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} - \frac{6cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} - \frac{6x*-e^{lg(sin(x))}cos(x)cos^{2}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin^{2}(x)} - \frac{6x*-0cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} - \frac{6x*-2cos(x)cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} - \frac{6x*-2cos(x)sin(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{3*2xcos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} + \frac{3x^{2}*-e^{lg(sin(x))}cos(x)cos^{3}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln^{2}{10}sin^{3}(x)} + \frac{3x^{2}*-2*0cos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} + \frac{3x^{2}*-3cos(x)cos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{4}(x)} + \frac{3x^{2}*-3cos^{2}(x)sin(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{4cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{4x*-e^{lg(sin(x))}cos(x)cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin(x)} - \frac{4x*-0cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} - \frac{4x*-cos(x)cos(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} - \frac{4x*-sin(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{6*-e^{lg(sin(x))}cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}} + \frac{6*-0}{e^{lg(sin(x))}ln^{2}{10}} - \frac{6cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} - \frac{6x*-e^{lg(sin(x))}cos(x)cos^{2}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln^{2}{10}sin^{2}(x)} - \frac{6x*-2*0cos^{2}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{2}(x)} - \frac{6x*-2cos(x)cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{6x*-2cos(x)sin(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{2*2xcos^{3}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} + \frac{2x^{2}*-e^{lg(sin(x))}cos(x)cos^{3}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin^{3}(x)} + \frac{2x^{2}*-0cos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} + \frac{2x^{2}*-3cos(x)cos^{3}(x)}{e^{lg(sin(x))}ln{10}sin^{4}(x)} + \frac{2x^{2}*-3cos^{2}(x)sin(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} + \frac{3*2xcos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} + \frac{3x^{2}*-e^{lg(sin(x))}cos(x)cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln^{2}{10}sin(x)} + \frac{3x^{2}*-2*0cos(x)}{e^{lg(sin(x))}ln^{3}{10}sin(x)} + \frac{3x^{2}*-cos(x)cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{3x^{2}*-sin(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} + \frac{2xcos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} + \frac{x^{2}*-e^{lg(sin(x))}cos(x)cos^{3}(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln^{3}{10}sin^{3}(x)} + \frac{x^{2}*-3*0cos^{3}(x)}{e^{lg(sin(x))}ln^{4}{10}sin^{3}(x)} + \frac{x^{2}*-3cos(x)cos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{4}(x)} + \frac{x^{2}*-3cos^{2}(x)sin(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} + \frac{2*2xcos(x)}{e^{lg(sin(x))}ln{10}sin(x)} + \frac{2x^{2}*-e^{lg(sin(x))}cos(x)cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}sin(x)} + \frac{2x^{2}*-0cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} + \frac{2x^{2}*-cos(x)cos(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{2x^{2}*-sin(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{6}{e^{lg(sin(x))}ln{10}} - \frac{6x*-e^{lg(sin(x))}cos(x)}{e^{{lg(sin(x))}*{2}}ln{10}(sin(x))ln{10}} - \frac{6x*-0}{e^{lg(sin(x))}ln^{2}{10}}\\=&\frac{-8cos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} - \frac{24cos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} - \frac{24cos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} - \frac{16cos^{3}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} - \frac{12cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} - \frac{16cos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{12cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} - \frac{12}{e^{lg(sin(x))}ln{10}} + \frac{2xcos^{4}(x)}{e^{lg(sin(x))}ln^{4}{10}sin^{4}(x)} + \frac{12xcos^{4}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{4}(x)} + \frac{22xcos^{4}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{4}(x)} + \frac{12xcos^{2}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{2}(x)} + \frac{12xcos^{4}(x)}{e^{lg(sin(x))}ln{10}sin^{4}(x)} + \frac{28xcos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} + \frac{16xcos^{3}(x)}{e^{lg(sin(x))}ln{10}sin^{3}(x)} - \frac{11x^{2}cos^{4}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{4}(x)} + \frac{16xcos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{24xcos^{3}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{3}(x)} + \frac{16xcos(x)}{e^{lg(sin(x))}ln{10}sin(x)} - \frac{6x^{2}cos^{4}(x)}{e^{lg(sin(x))}ln{10}sin^{4}(x)} - \frac{14x^{2}cos^{2}(x)}{e^{lg(sin(x))}ln^{2}{10}sin^{2}(x)} - \frac{6x^{2}cos^{4}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{4}(x)} - \frac{8x^{2}cos^{2}(x)}{e^{lg(sin(x))}ln{10}sin^{2}(x)} + \frac{24xcos(x)}{e^{lg(sin(x))}ln^{2}{10}sin(x)} + \frac{8xcos^{3}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{3}(x)} - \frac{6x^{2}cos^{2}(x)}{e^{lg(sin(x))}ln^{3}{10}sin^{2}(x)} - \frac{x^{2}cos^{4}(x)}{e^{lg(sin(x))}ln^{4}{10}sin^{4}(x)} - \frac{3x^{2}}{e^{lg(sin(x))}ln^{2}{10}} + \frac{4x}{e^{lg(sin(x))}ln{10}} + \frac{6x}{e^{lg(sin(x))}ln^{2}{10}} - \frac{2x^{2}}{e^{lg(sin(x))}ln{10}}\\ \end{split}\end{equation} \]