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 t is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ (lg(t))(sin(t))(cos(t))(e^{t})(tt - 2t - 3)\ with\ respect\ to\ t：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = t^{2}e^{t}lg(t)sin(t)cos(t) - 2te^{t}lg(t)sin(t)cos(t) - 3e^{t}lg(t)sin(t)cos(t)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( t^{2}e^{t}lg(t)sin(t)cos(t) - 2te^{t}lg(t)sin(t)cos(t) - 3e^{t}lg(t)sin(t)cos(t)\right)}{dt}\\=&2te^{t}lg(t)sin(t)cos(t) + t^{2}e^{t}lg(t)sin(t)cos(t) + \frac{t^{2}e^{t}sin(t)cos(t)}{ln{10}(t)} + t^{2}e^{t}lg(t)cos(t)cos(t) + t^{2}e^{t}lg(t)sin(t)*-sin(t) - 2e^{t}lg(t)sin(t)cos(t) - 2te^{t}lg(t)sin(t)cos(t) - \frac{2te^{t}sin(t)cos(t)}{ln{10}(t)} - 2te^{t}lg(t)cos(t)cos(t) - 2te^{t}lg(t)sin(t)*-sin(t) - 3e^{t}lg(t)sin(t)cos(t) - \frac{3e^{t}sin(t)cos(t)}{ln{10}(t)} - 3e^{t}lg(t)cos(t)cos(t) - 3e^{t}lg(t)sin(t)*-sin(t)\\=&t^{2}e^{t}lg(t)sin(t)cos(t) + \frac{te^{t}sin(t)cos(t)}{ln{10}} + t^{2}e^{t}lg(t)cos^{2}(t) - t^{2}e^{t}lg(t)sin^{2}(t) - 5e^{t}lg(t)sin(t)cos(t) - \frac{2e^{t}sin(t)cos(t)}{ln{10}} - 2te^{t}lg(t)cos^{2}(t) + 2te^{t}lg(t)sin^{2}(t) - \frac{3e^{t}sin(t)cos(t)}{tln{10}} - 3e^{t}lg(t)cos^{2}(t) + 3e^{t}lg(t)sin^{2}(t)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( t^{2}e^{t}lg(t)sin(t)cos(t) + \frac{te^{t}sin(t)cos(t)}{ln{10}} + t^{2}e^{t}lg(t)cos^{2}(t) - t^{2}e^{t}lg(t)sin^{2}(t) - 5e^{t}lg(t)sin(t)cos(t) - \frac{2e^{t}sin(t)cos(t)}{ln{10}} - 2te^{t}lg(t)cos^{2}(t) + 2te^{t}lg(t)sin^{2}(t) - \frac{3e^{t}sin(t)cos(t)}{tln{10}} - 3e^{t}lg(t)cos^{2}(t) + 3e^{t}lg(t)sin^{2}(t)\right)}{dt}\\=&2te^{t}lg(t)sin(t)cos(t) + t^{2}e^{t}lg(t)sin(t)cos(t) + \frac{t^{2}e^{t}sin(t)cos(t)}{ln{10}(t)} + t^{2}e^{t}lg(t)cos(t)cos(t) + t^{2}e^{t}lg(t)sin(t)*-sin(t) + \frac{e^{t}sin(t)cos(t)}{ln{10}} + \frac{te^{t}sin(t)cos(t)}{ln{10}} + \frac{te^{t}*-0sin(t)cos(t)}{ln^{2}{10}} + \frac{te^{t}cos(t)cos(t)}{ln{10}} + \frac{te^{t}sin(t)*-sin(t)}{ln{10}} + 2te^{t}lg(t)cos^{2}(t) + t^{2}e^{t}lg(t)cos^{2}(t) + \frac{t^{2}e^{t}cos^{2}(t)}{ln{10}(t)} + t^{2}e^{t}lg(t)*-2cos(t)sin(t) - 2te^{t}lg(t)sin^{2}(t) - t^{2}e^{t}lg(t)sin^{2}(t) - \frac{t^{2}e^{t}sin^{2}(t)}{ln{10}(t)} - t^{2}e^{t}lg(t)*2sin(t)cos(t) - 5e^{t}lg(t)sin(t)cos(t) - \frac{5e^{t}sin(t)cos(t)}{ln{10}(t)} - 5e^{t}lg(t)cos(t)cos(t) - 5e^{t}lg(t)sin(t)*-sin(t) - \frac{2e^{t}sin(t)cos(t)}{ln{10}} - \frac{2e^{t}*-0sin(t)cos(t)}{ln^{2}{10}} - \frac{2e^{t}cos(t)cos(t)}{ln{10}} - \frac{2e^{t}sin(t)*-sin(t)}{ln{10}} - 2e^{t}lg(t)cos^{2}(t) - 2te^{t}lg(t)cos^{2}(t) - \frac{2te^{t}cos^{2}(t)}{ln{10}(t)} - 2te^{t}lg(t)*-2cos(t)sin(t) + 2e^{t}lg(t)sin^{2}(t) + 2te^{t}lg(t)sin^{2}(t) + \frac{2te^{t}sin^{2}(t)}{ln{10}(t)} + 2te^{t}lg(t)*2sin(t)cos(t) - \frac{3*-e^{t}sin(t)cos(t)}{t^{2}ln{10}} - \frac{3e^{t}sin(t)cos(t)}{tln{10}} - \frac{3e^{t}*-0sin(t)cos(t)}{tln^{2}{10}} - \frac{3e^{t}cos(t)cos(t)}{tln{10}} - \frac{3e^{t}sin(t)*-sin(t)}{tln{10}} - 3e^{t}lg(t)cos^{2}(t) - \frac{3e^{t}cos^{2}(t)}{ln{10}(t)} - 3e^{t}lg(t)*-2cos(t)sin(t) + 3e^{t}lg(t)sin^{2}(t) + \frac{3e^{t}sin^{2}(t)}{ln{10}(t)} + 3e^{t}lg(t)*2sin(t)cos(t)\\=&10te^{t}lg(t)sin(t)cos(t) - 3t^{2}e^{t}lg(t)sin(t)cos(t) + \frac{2te^{t}sin(t)cos(t)}{ln{10}} + 2t^{2}e^{t}lg(t)cos^{2}(t) - 2t^{2}e^{t}lg(t)sin^{2}(t) - \frac{e^{t}sin(t)cos(t)}{ln{10}} + \frac{2te^{t}cos^{2}(t)}{ln{10}} - \frac{8e^{t}sin(t)cos(t)}{tln{10}} + 7e^{t}lg(t)sin(t)cos(t) + \frac{3e^{t}sin(t)cos(t)}{t^{2}ln{10}} - 10e^{t}lg(t)cos^{2}(t) + 10e^{t}lg(t)sin^{2}(t) - \frac{4e^{t}cos^{2}(t)}{ln{10}} + \frac{4e^{t}sin^{2}(t)}{ln{10}} - \frac{2te^{t}sin^{2}(t)}{ln{10}} - \frac{6e^{t}cos^{2}(t)}{tln{10}} + \frac{6e^{t}sin^{2}(t)}{tln{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 10te^{t}lg(t)sin(t)cos(t) - 3t^{2}e^{t}lg(t)sin(t)cos(t) + \frac{2te^{t}sin(t)cos(t)}{ln{10}} + 2t^{2}e^{t}lg(t)cos^{2}(t) - 2t^{2}e^{t}lg(t)sin^{2}(t) - \frac{e^{t}sin(t)cos(t)}{ln{10}} + \frac{2te^{t}cos^{2}(t)}{ln{10}} - \frac{8e^{t}sin(t)cos(t)}{tln{10}} + 7e^{t}lg(t)sin(t)cos(t) + \frac{3e^{t}sin(t)cos(t)}{t^{2}ln{10}} - 10e^{t}lg(t)cos^{2}(t) + 10e^{t}lg(t)sin^{2}(t) - \frac{4e^{t}cos^{2}(t)}{ln{10}} + \frac{4e^{t}sin^{2}(t)}{ln{10}} - \frac{2te^{t}sin^{2}(t)}{ln{10}} - \frac{6e^{t}cos^{2}(t)}{tln{10}} + \frac{6e^{t}sin^{2}(t)}{tln{10}}\right)}{dt}\\=&10e^{t}lg(t)sin(t)cos(t) + 10te^{t}lg(t)sin(t)cos(t) + \frac{10te^{t}sin(t)cos(t)}{ln{10}(t)} + 10te^{t}lg(t)cos(t)cos(t) + 10te^{t}lg(t)sin(t)*-sin(t) - 3*2te^{t}lg(t)sin(t)cos(t) - 3t^{2}e^{t}lg(t)sin(t)cos(t) - \frac{3t^{2}e^{t}sin(t)cos(t)}{ln{10}(t)} - 3t^{2}e^{t}lg(t)cos(t)cos(t) - 3t^{2}e^{t}lg(t)sin(t)*-sin(t) + \frac{2e^{t}sin(t)cos(t)}{ln{10}} + \frac{2te^{t}sin(t)cos(t)}{ln{10}} + \frac{2te^{t}*-0sin(t)cos(t)}{ln^{2}{10}} + \frac{2te^{t}cos(t)cos(t)}{ln{10}} + \frac{2te^{t}sin(t)*-sin(t)}{ln{10}} + 2*2te^{t}lg(t)cos^{2}(t) + 2t^{2}e^{t}lg(t)cos^{2}(t) + \frac{2t^{2}e^{t}cos^{2}(t)}{ln{10}(t)} + 2t^{2}e^{t}lg(t)*-2cos(t)sin(t) - 2*2te^{t}lg(t)sin^{2}(t) - 2t^{2}e^{t}lg(t)sin^{2}(t) - \frac{2t^{2}e^{t}sin^{2}(t)}{ln{10}(t)} - 2t^{2}e^{t}lg(t)*2sin(t)cos(t) - \frac{e^{t}sin(t)cos(t)}{ln{10}} - \frac{e^{t}*-0sin(t)cos(t)}{ln^{2}{10}} - \frac{e^{t}cos(t)cos(t)}{ln{10}} - \frac{e^{t}sin(t)*-sin(t)}{ln{10}} + \frac{2e^{t}cos^{2}(t)}{ln{10}} + \frac{2te^{t}cos^{2}(t)}{ln{10}} + \frac{2te^{t}*-0cos^{2}(t)}{ln^{2}{10}} + \frac{2te^{t}*-2cos(t)sin(t)}{ln{10}} - \frac{8*-e^{t}sin(t)cos(t)}{t^{2}ln{10}} - \frac{8e^{t}sin(t)cos(t)}{tln{10}} - \frac{8e^{t}*-0sin(t)cos(t)}{tln^{2}{10}} - \frac{8e^{t}cos(t)cos(t)}{tln{10}} - \frac{8e^{t}sin(t)*-sin(t)}{tln{10}} + 7e^{t}lg(t)sin(t)cos(t) + \frac{7e^{t}sin(t)cos(t)}{ln{10}(t)} + 7e^{t}lg(t)cos(t)cos(t) + 7e^{t}lg(t)sin(t)*-sin(t) + \frac{3*-2e^{t}sin(t)cos(t)}{t^{3}ln{10}} + \frac{3e^{t}sin(t)cos(t)}{t^{2}ln{10}} + \frac{3e^{t}*-0sin(t)cos(t)}{t^{2}ln^{2}{10}} + \frac{3e^{t}cos(t)cos(t)}{t^{2}ln{10}} + \frac{3e^{t}sin(t)*-sin(t)}{t^{2}ln{10}} - 10e^{t}lg(t)cos^{2}(t) - \frac{10e^{t}cos^{2}(t)}{ln{10}(t)} - 10e^{t}lg(t)*-2cos(t)sin(t) + 10e^{t}lg(t)sin^{2}(t) + \frac{10e^{t}sin^{2}(t)}{ln{10}(t)} + 10e^{t}lg(t)*2sin(t)cos(t) - \frac{4e^{t}cos^{2}(t)}{ln{10}} - \frac{4e^{t}*-0cos^{2}(t)}{ln^{2}{10}} - \frac{4e^{t}*-2cos(t)sin(t)}{ln{10}} + \frac{4e^{t}sin^{2}(t)}{ln{10}} + \frac{4e^{t}*-0sin^{2}(t)}{ln^{2}{10}} + \frac{4e^{t}*2sin(t)cos(t)}{ln{10}} - \frac{2e^{t}sin^{2}(t)}{ln{10}} - \frac{2te^{t}sin^{2}(t)}{ln{10}} - \frac{2te^{t}*-0sin^{2}(t)}{ln^{2}{10}} - \frac{2te^{t}*2sin(t)cos(t)}{ln{10}} - \frac{6*-e^{t}cos^{2}(t)}{t^{2}ln{10}} - \frac{6e^{t}cos^{2}(t)}{tln{10}} - \frac{6e^{t}*-0cos^{2}(t)}{tln^{2}{10}} - \frac{6e^{t}*-2cos(t)sin(t)}{tln{10}} + \frac{6*-e^{t}sin^{2}(t)}{t^{2}ln{10}} + \frac{6e^{t}sin^{2}(t)}{tln{10}} + \frac{6e^{t}*-0sin^{2}(t)}{tln^{2}{10}} + \frac{6e^{t}*2sin(t)cos(t)}{tln{10}}\\=&57e^{t}lg(t)sin(t)cos(t) + 4te^{t}lg(t)sin(t)cos(t) + \frac{27e^{t}sin(t)cos(t)}{ln{10}} + 14te^{t}lg(t)cos^{2}(t) - 11t^{2}e^{t}lg(t)sin(t)cos(t) - 14te^{t}lg(t)sin^{2}(t) - \frac{9te^{t}sin(t)cos(t)}{ln{10}} - t^{2}e^{t}lg(t)cos^{2}(t) + t^{2}e^{t}lg(t)sin^{2}(t) + \frac{6te^{t}cos^{2}(t)}{ln{10}} + \frac{11e^{t}sin(t)cos(t)}{t^{2}ln{10}} - \frac{3e^{t}cos^{2}(t)}{ln{10}} + \frac{3e^{t}sin^{2}(t)}{ln{10}} - \frac{24e^{t}cos^{2}(t)}{tln{10}} + \frac{23e^{t}sin(t)cos(t)}{tln{10}} - \frac{6e^{t}sin(t)cos(t)}{t^{3}ln{10}} + \frac{9e^{t}cos^{2}(t)}{t^{2}ln{10}} + \frac{24e^{t}sin^{2}(t)}{tln{10}} - 3e^{t}lg(t)cos^{2}(t) + 3e^{t}lg(t)sin^{2}(t) - \frac{6te^{t}sin^{2}(t)}{ln{10}} - \frac{9e^{t}sin^{2}(t)}{t^{2}ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 57e^{t}lg(t)sin(t)cos(t) + 4te^{t}lg(t)sin(t)cos(t) + \frac{27e^{t}sin(t)cos(t)}{ln{10}} + 14te^{t}lg(t)cos^{2}(t) - 11t^{2}e^{t}lg(t)sin(t)cos(t) - 14te^{t}lg(t)sin^{2}(t) - \frac{9te^{t}sin(t)cos(t)}{ln{10}} - t^{2}e^{t}lg(t)cos^{2}(t) + t^{2}e^{t}lg(t)sin^{2}(t) + \frac{6te^{t}cos^{2}(t)}{ln{10}} + \frac{11e^{t}sin(t)cos(t)}{t^{2}ln{10}} - \frac{3e^{t}cos^{2}(t)}{ln{10}} + \frac{3e^{t}sin^{2}(t)}{ln{10}} - \frac{24e^{t}cos^{2}(t)}{tln{10}} + \frac{23e^{t}sin(t)cos(t)}{tln{10}} - \frac{6e^{t}sin(t)cos(t)}{t^{3}ln{10}} + \frac{9e^{t}cos^{2}(t)}{t^{2}ln{10}} + \frac{24e^{t}sin^{2}(t)}{tln{10}} - 3e^{t}lg(t)cos^{2}(t) + 3e^{t}lg(t)sin^{2}(t) - \frac{6te^{t}sin^{2}(t)}{ln{10}} - \frac{9e^{t}sin^{2}(t)}{t^{2}ln{10}}\right)}{dt}\\=&57e^{t}lg(t)sin(t)cos(t) + \frac{57e^{t}sin(t)cos(t)}{ln{10}(t)} + 57e^{t}lg(t)cos(t)cos(t) + 57e^{t}lg(t)sin(t)*-sin(t) + 4e^{t}lg(t)sin(t)cos(t) + 4te^{t}lg(t)sin(t)cos(t) + \frac{4te^{t}sin(t)cos(t)}{ln{10}(t)} + 4te^{t}lg(t)cos(t)cos(t) + 4te^{t}lg(t)sin(t)*-sin(t) + \frac{27e^{t}sin(t)cos(t)}{ln{10}} + \frac{27e^{t}*-0sin(t)cos(t)}{ln^{2}{10}} + \frac{27e^{t}cos(t)cos(t)}{ln{10}} + \frac{27e^{t}sin(t)*-sin(t)}{ln{10}} + 14e^{t}lg(t)cos^{2}(t) + 14te^{t}lg(t)cos^{2}(t) + \frac{14te^{t}cos^{2}(t)}{ln{10}(t)} + 14te^{t}lg(t)*-2cos(t)sin(t) - 11*2te^{t}lg(t)sin(t)cos(t) - 11t^{2}e^{t}lg(t)sin(t)cos(t) - \frac{11t^{2}e^{t}sin(t)cos(t)}{ln{10}(t)} - 11t^{2}e^{t}lg(t)cos(t)cos(t) - 11t^{2}e^{t}lg(t)sin(t)*-sin(t) - 14e^{t}lg(t)sin^{2}(t) - 14te^{t}lg(t)sin^{2}(t) - \frac{14te^{t}sin^{2}(t)}{ln{10}(t)} - 14te^{t}lg(t)*2sin(t)cos(t) - \frac{9e^{t}sin(t)cos(t)}{ln{10}} - \frac{9te^{t}sin(t)cos(t)}{ln{10}} - \frac{9te^{t}*-0sin(t)cos(t)}{ln^{2}{10}} - \frac{9te^{t}cos(t)cos(t)}{ln{10}} - \frac{9te^{t}sin(t)*-sin(t)}{ln{10}} - 2te^{t}lg(t)cos^{2}(t) - t^{2}e^{t}lg(t)cos^{2}(t) - \frac{t^{2}e^{t}cos^{2}(t)}{ln{10}(t)} - t^{2}e^{t}lg(t)*-2cos(t)sin(t) + 2te^{t}lg(t)sin^{2}(t) + t^{2}e^{t}lg(t)sin^{2}(t) + \frac{t^{2}e^{t}sin^{2}(t)}{ln{10}(t)} + t^{2}e^{t}lg(t)*2sin(t)cos(t) + \frac{6e^{t}cos^{2}(t)}{ln{10}} + \frac{6te^{t}cos^{2}(t)}{ln{10}} + \frac{6te^{t}*-0cos^{2}(t)}{ln^{2}{10}} + \frac{6te^{t}*-2cos(t)sin(t)}{ln{10}} + \frac{11*-2e^{t}sin(t)cos(t)}{t^{3}ln{10}} + \frac{11e^{t}sin(t)cos(t)}{t^{2}ln{10}} + \frac{11e^{t}*-0sin(t)cos(t)}{t^{2}ln^{2}{10}} + \frac{11e^{t}cos(t)cos(t)}{t^{2}ln{10}} + \frac{11e^{t}sin(t)*-sin(t)}{t^{2}ln{10}} - \frac{3e^{t}cos^{2}(t)}{ln{10}} - \frac{3e^{t}*-0cos^{2}(t)}{ln^{2}{10}} - \frac{3e^{t}*-2cos(t)sin(t)}{ln{10}} + \frac{3e^{t}sin^{2}(t)}{ln{10}} + \frac{3e^{t}*-0sin^{2}(t)}{ln^{2}{10}} + \frac{3e^{t}*2sin(t)cos(t)}{ln{10}} - \frac{24*-e^{t}cos^{2}(t)}{t^{2}ln{10}} - \frac{24e^{t}cos^{2}(t)}{tln{10}} - \frac{24e^{t}*-0cos^{2}(t)}{tln^{2}{10}} - \frac{24e^{t}*-2cos(t)sin(t)}{tln{10}} + \frac{23*-e^{t}sin(t)cos(t)}{t^{2}ln{10}} + \frac{23e^{t}sin(t)cos(t)}{tln{10}} + \frac{23e^{t}*-0sin(t)cos(t)}{tln^{2}{10}} + \frac{23e^{t}cos(t)cos(t)}{tln{10}} + \frac{23e^{t}sin(t)*-sin(t)}{tln{10}} - \frac{6*-3e^{t}sin(t)cos(t)}{t^{4}ln{10}} - \frac{6e^{t}sin(t)cos(t)}{t^{3}ln{10}} - \frac{6e^{t}*-0sin(t)cos(t)}{t^{3}ln^{2}{10}} - \frac{6e^{t}cos(t)cos(t)}{t^{3}ln{10}} - \frac{6e^{t}sin(t)*-sin(t)}{t^{3}ln{10}} + \frac{9*-2e^{t}cos^{2}(t)}{t^{3}ln{10}} + \frac{9e^{t}cos^{2}(t)}{t^{2}ln{10}} + \frac{9e^{t}*-0cos^{2}(t)}{t^{2}ln^{2}{10}} + \frac{9e^{t}*-2cos(t)sin(t)}{t^{2}ln{10}} + \frac{24*-e^{t}sin^{2}(t)}{t^{2}ln{10}} + \frac{24e^{t}sin^{2}(t)}{tln{10}} + \frac{24e^{t}*-0sin^{2}(t)}{tln^{2}{10}} + \frac{24e^{t}*2sin(t)cos(t)}{tln{10}} - 3e^{t}lg(t)cos^{2}(t) - \frac{3e^{t}cos^{2}(t)}{ln{10}(t)} - 3e^{t}lg(t)*-2cos(t)sin(t) + 3e^{t}lg(t)sin^{2}(t) + \frac{3e^{t}sin^{2}(t)}{ln{10}(t)} + 3e^{t}lg(t)*2sin(t)cos(t) - \frac{6e^{t}sin^{2}(t)}{ln{10}} - \frac{6te^{t}sin^{2}(t)}{ln{10}} - \frac{6te^{t}*-0sin^{2}(t)}{ln^{2}{10}} - \frac{6te^{t}*2sin(t)cos(t)}{ln{10}} - \frac{9*-2e^{t}sin^{2}(t)}{t^{3}ln{10}} - \frac{9e^{t}sin^{2}(t)}{t^{2}ln{10}} - \frac{9e^{t}*-0sin^{2}(t)}{t^{2}ln^{2}{10}} - \frac{9e^{t}*2sin(t)cos(t)}{t^{2}ln{10}}\\=&73e^{t}lg(t)sin(t)cos(t) + \frac{176e^{t}sin(t)cos(t)}{tln{10}} + 68e^{t}lg(t)cos^{2}(t) - 68e^{t}lg(t)sin^{2}(t) - 74te^{t}lg(t)sin(t)cos(t) + \frac{34e^{t}sin(t)cos(t)}{ln{10}} + 16te^{t}lg(t)cos^{2}(t) - 7t^{2}e^{t}lg(t)sin(t)cos(t) + \frac{44e^{t}cos^{2}(t)}{ln{10}} - \frac{44e^{t}sin^{2}(t)}{ln{10}} - 12t^{2}e^{t}lg(t)cos^{2}(t) - 16te^{t}lg(t)sin^{2}(t) - \frac{44te^{t}sin(t)cos(t)}{ln{10}} + 12t^{2}e^{t}lg(t)sin^{2}(t) - \frac{4te^{t}cos^{2}(t)}{ln{10}} - \frac{28e^{t}sin(t)cos(t)}{t^{3}ln{10}} + \frac{44e^{t}cos^{2}(t)}{t^{2}ln{10}} - \frac{48e^{t}sin(t)cos(t)}{t^{2}ln{10}} + \frac{18e^{t}sin(t)cos(t)}{t^{4}ln{10}} - \frac{4e^{t}cos^{2}(t)}{tln{10}} - \frac{44e^{t}sin^{2}(t)}{t^{2}ln{10}} - \frac{24e^{t}cos^{2}(t)}{t^{3}ln{10}} + \frac{4e^{t}sin^{2}(t)}{tln{10}} + \frac{4te^{t}sin^{2}(t)}{ln{10}} + \frac{24e^{t}sin^{2}(t)}{t^{3}ln{10}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ (lg(t))(sin(t))(cos(t))(e^{t})(tt - 2t - 3)\ with\ respect\ to\ t：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = t^{2}e^{t}lg(t)sin(t)cos(t) - 2te^{t}lg(t)sin(t)cos(t) - 3e^{t}lg(t)sin(t)cos(t)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( t^{2}e^{t}lg(t)sin(t)cos(t) - 2te^{t}lg(t)sin(t)cos(t) - 3e^{t}lg(t)sin(t)cos(t)\right)}{dt}\\=&2te^{t}lg(t)sin(t)cos(t) + t^{2}e^{t}lg(t)sin(t)cos(t) + \frac{t^{2}e^{t}sin(t)cos(t)}{ln{10}(t)} + t^{2}e^{t}lg(t)cos(t)cos(t) + t^{2}e^{t}lg(t)sin(t)*-sin(t) - 2e^{t}lg(t)sin(t)cos(t) - 2te^{t}lg(t)sin(t)cos(t) - \frac{2te^{t}sin(t)cos(t)}{ln{10}(t)} - 2te^{t}lg(t)cos(t)cos(t) - 2te^{t}lg(t)sin(t)*-sin(t) - 3e^{t}lg(t)sin(t)cos(t) - \frac{3e^{t}sin(t)cos(t)}{ln{10}(t)} - 3e^{t}lg(t)cos(t)cos(t) - 3e^{t}lg(t)sin(t)*-sin(t)\\=&t^{2}e^{t}lg(t)sin(t)cos(t) + \frac{te^{t}sin(t)cos(t)}{ln{10}} + t^{2}e^{t}lg(t)cos^{2}(t) - t^{2}e^{t}lg(t)sin^{2}(t) - 5e^{t}lg(t)sin(t)cos(t) - \frac{2e^{t}sin(t)cos(t)}{ln{10}} - 2te^{t}lg(t)cos^{2}(t) + 2te^{t}lg(t)sin^{2}(t) - \frac{3e^{t}sin(t)cos(t)}{tln{10}} - 3e^{t}lg(t)cos^{2}(t) + 3e^{t}lg(t)sin^{2}(t)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( t^{2}e^{t}lg(t)sin(t)cos(t) + \frac{te^{t}sin(t)cos(t)}{ln{10}} + t^{2}e^{t}lg(t)cos^{2}(t) - t^{2}e^{t}lg(t)sin^{2}(t) - 5e^{t}lg(t)sin(t)cos(t) - \frac{2e^{t}sin(t)cos(t)}{ln{10}} - 2te^{t}lg(t)cos^{2}(t) + 2te^{t}lg(t)sin^{2}(t) - \frac{3e^{t}sin(t)cos(t)}{tln{10}} - 3e^{t}lg(t)cos^{2}(t) + 3e^{t}lg(t)sin^{2}(t)\right)}{dt}\\=&2te^{t}lg(t)sin(t)cos(t) + t^{2}e^{t}lg(t)sin(t)cos(t) + \frac{t^{2}e^{t}sin(t)cos(t)}{ln{10}(t)} + t^{2}e^{t}lg(t)cos(t)cos(t) + t^{2}e^{t}lg(t)sin(t)*-sin(t) + \frac{e^{t}sin(t)cos(t)}{ln{10}} + \frac{te^{t}sin(t)cos(t)}{ln{10}} + \frac{te^{t}*-0sin(t)cos(t)}{ln^{2}{10}} + \frac{te^{t}cos(t)cos(t)}{ln{10}} + \frac{te^{t}sin(t)*-sin(t)}{ln{10}} + 2te^{t}lg(t)cos^{2}(t) + t^{2}e^{t}lg(t)cos^{2}(t) + \frac{t^{2}e^{t}cos^{2}(t)}{ln{10}(t)} + t^{2}e^{t}lg(t)*-2cos(t)sin(t) - 2te^{t}lg(t)sin^{2}(t) - t^{2}e^{t}lg(t)sin^{2}(t) - \frac{t^{2}e^{t}sin^{2}(t)}{ln{10}(t)} - t^{2}e^{t}lg(t)*2sin(t)cos(t) - 5e^{t}lg(t)sin(t)cos(t) - \frac{5e^{t}sin(t)cos(t)}{ln{10}(t)} - 5e^{t}lg(t)cos(t)cos(t) - 5e^{t}lg(t)sin(t)*-sin(t) - \frac{2e^{t}sin(t)cos(t)}{ln{10}} - \frac{2e^{t}*-0sin(t)cos(t)}{ln^{2}{10}} - \frac{2e^{t}cos(t)cos(t)}{ln{10}} - \frac{2e^{t}sin(t)*-sin(t)}{ln{10}} - 2e^{t}lg(t)cos^{2}(t) - 2te^{t}lg(t)cos^{2}(t) - \frac{2te^{t}cos^{2}(t)}{ln{10}(t)} - 2te^{t}lg(t)*-2cos(t)sin(t) + 2e^{t}lg(t)sin^{2}(t) + 2te^{t}lg(t)sin^{2}(t) + \frac{2te^{t}sin^{2}(t)}{ln{10}(t)} + 2te^{t}lg(t)*2sin(t)cos(t) - \frac{3*-e^{t}sin(t)cos(t)}{t^{2}ln{10}} - \frac{3e^{t}sin(t)cos(t)}{tln{10}} - \frac{3e^{t}*-0sin(t)cos(t)}{tln^{2}{10}} - \frac{3e^{t}cos(t)cos(t)}{tln{10}} - \frac{3e^{t}sin(t)*-sin(t)}{tln{10}} - 3e^{t}lg(t)cos^{2}(t) - \frac{3e^{t}cos^{2}(t)}{ln{10}(t)} - 3e^{t}lg(t)*-2cos(t)sin(t) + 3e^{t}lg(t)sin^{2}(t) + \frac{3e^{t}sin^{2}(t)}{ln{10}(t)} + 3e^{t}lg(t)*2sin(t)cos(t)\\=&10te^{t}lg(t)sin(t)cos(t) - 3t^{2}e^{t}lg(t)sin(t)cos(t) + \frac{2te^{t}sin(t)cos(t)}{ln{10}} + 2t^{2}e^{t}lg(t)cos^{2}(t) - 2t^{2}e^{t}lg(t)sin^{2}(t) - \frac{e^{t}sin(t)cos(t)}{ln{10}} + \frac{2te^{t}cos^{2}(t)}{ln{10}} - \frac{8e^{t}sin(t)cos(t)}{tln{10}} + 7e^{t}lg(t)sin(t)cos(t) + \frac{3e^{t}sin(t)cos(t)}{t^{2}ln{10}} - 10e^{t}lg(t)cos^{2}(t) + 10e^{t}lg(t)sin^{2}(t) - \frac{4e^{t}cos^{2}(t)}{ln{10}} + \frac{4e^{t}sin^{2}(t)}{ln{10}} - \frac{2te^{t}sin^{2}(t)}{ln{10}} - \frac{6e^{t}cos^{2}(t)}{tln{10}} + \frac{6e^{t}sin^{2}(t)}{tln{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 10te^{t}lg(t)sin(t)cos(t) - 3t^{2}e^{t}lg(t)sin(t)cos(t) + \frac{2te^{t}sin(t)cos(t)}{ln{10}} + 2t^{2}e^{t}lg(t)cos^{2}(t) - 2t^{2}e^{t}lg(t)sin^{2}(t) - \frac{e^{t}sin(t)cos(t)}{ln{10}} + \frac{2te^{t}cos^{2}(t)}{ln{10}} - \frac{8e^{t}sin(t)cos(t)}{tln{10}} + 7e^{t}lg(t)sin(t)cos(t) + \frac{3e^{t}sin(t)cos(t)}{t^{2}ln{10}} - 10e^{t}lg(t)cos^{2}(t) + 10e^{t}lg(t)sin^{2}(t) - \frac{4e^{t}cos^{2}(t)}{ln{10}} + \frac{4e^{t}sin^{2}(t)}{ln{10}} - \frac{2te^{t}sin^{2}(t)}{ln{10}} - \frac{6e^{t}cos^{2}(t)}{tln{10}} + \frac{6e^{t}sin^{2}(t)}{tln{10}}\right)}{dt}\\=&10e^{t}lg(t)sin(t)cos(t) + 10te^{t}lg(t)sin(t)cos(t) + \frac{10te^{t}sin(t)cos(t)}{ln{10}(t)} + 10te^{t}lg(t)cos(t)cos(t) + 10te^{t}lg(t)sin(t)*-sin(t) - 3*2te^{t}lg(t)sin(t)cos(t) - 3t^{2}e^{t}lg(t)sin(t)cos(t) - \frac{3t^{2}e^{t}sin(t)cos(t)}{ln{10}(t)} - 3t^{2}e^{t}lg(t)cos(t)cos(t) - 3t^{2}e^{t}lg(t)sin(t)*-sin(t) + \frac{2e^{t}sin(t)cos(t)}{ln{10}} + \frac{2te^{t}sin(t)cos(t)}{ln{10}} + \frac{2te^{t}*-0sin(t)cos(t)}{ln^{2}{10}} + \frac{2te^{t}cos(t)cos(t)}{ln{10}} + \frac{2te^{t}sin(t)*-sin(t)}{ln{10}} + 2*2te^{t}lg(t)cos^{2}(t) + 2t^{2}e^{t}lg(t)cos^{2}(t) + \frac{2t^{2}e^{t}cos^{2}(t)}{ln{10}(t)} + 2t^{2}e^{t}lg(t)*-2cos(t)sin(t) - 2*2te^{t}lg(t)sin^{2}(t) - 2t^{2}e^{t}lg(t)sin^{2}(t) - \frac{2t^{2}e^{t}sin^{2}(t)}{ln{10}(t)} - 2t^{2}e^{t}lg(t)*2sin(t)cos(t) - \frac{e^{t}sin(t)cos(t)}{ln{10}} - \frac{e^{t}*-0sin(t)cos(t)}{ln^{2}{10}} - \frac{e^{t}cos(t)cos(t)}{ln{10}} - \frac{e^{t}sin(t)*-sin(t)}{ln{10}} + \frac{2e^{t}cos^{2}(t)}{ln{10}} + \frac{2te^{t}cos^{2}(t)}{ln{10}} + \frac{2te^{t}*-0cos^{2}(t)}{ln^{2}{10}} + \frac{2te^{t}*-2cos(t)sin(t)}{ln{10}} - \frac{8*-e^{t}sin(t)cos(t)}{t^{2}ln{10}} - \frac{8e^{t}sin(t)cos(t)}{tln{10}} - \frac{8e^{t}*-0sin(t)cos(t)}{tln^{2}{10}} - \frac{8e^{t}cos(t)cos(t)}{tln{10}} - \frac{8e^{t}sin(t)*-sin(t)}{tln{10}} + 7e^{t}lg(t)sin(t)cos(t) + \frac{7e^{t}sin(t)cos(t)}{ln{10}(t)} + 7e^{t}lg(t)cos(t)cos(t) + 7e^{t}lg(t)sin(t)*-sin(t) + \frac{3*-2e^{t}sin(t)cos(t)}{t^{3}ln{10}} + \frac{3e^{t}sin(t)cos(t)}{t^{2}ln{10}} + \frac{3e^{t}*-0sin(t)cos(t)}{t^{2}ln^{2}{10}} + \frac{3e^{t}cos(t)cos(t)}{t^{2}ln{10}} + \frac{3e^{t}sin(t)*-sin(t)}{t^{2}ln{10}} - 10e^{t}lg(t)cos^{2}(t) - \frac{10e^{t}cos^{2}(t)}{ln{10}(t)} - 10e^{t}lg(t)*-2cos(t)sin(t) + 10e^{t}lg(t)sin^{2}(t) + \frac{10e^{t}sin^{2}(t)}{ln{10}(t)} + 10e^{t}lg(t)*2sin(t)cos(t) - \frac{4e^{t}cos^{2}(t)}{ln{10}} - \frac{4e^{t}*-0cos^{2}(t)}{ln^{2}{10}} - \frac{4e^{t}*-2cos(t)sin(t)}{ln{10}} + \frac{4e^{t}sin^{2}(t)}{ln{10}} + \frac{4e^{t}*-0sin^{2}(t)}{ln^{2}{10}} + \frac{4e^{t}*2sin(t)cos(t)}{ln{10}} - \frac{2e^{t}sin^{2}(t)}{ln{10}} - \frac{2te^{t}sin^{2}(t)}{ln{10}} - \frac{2te^{t}*-0sin^{2}(t)}{ln^{2}{10}} - \frac{2te^{t}*2sin(t)cos(t)}{ln{10}} - \frac{6*-e^{t}cos^{2}(t)}{t^{2}ln{10}} - \frac{6e^{t}cos^{2}(t)}{tln{10}} - \frac{6e^{t}*-0cos^{2}(t)}{tln^{2}{10}} - \frac{6e^{t}*-2cos(t)sin(t)}{tln{10}} + \frac{6*-e^{t}sin^{2}(t)}{t^{2}ln{10}} + \frac{6e^{t}sin^{2}(t)}{tln{10}} + \frac{6e^{t}*-0sin^{2}(t)}{tln^{2}{10}} + \frac{6e^{t}*2sin(t)cos(t)}{tln{10}}\\=&57e^{t}lg(t)sin(t)cos(t) + 4te^{t}lg(t)sin(t)cos(t) + \frac{27e^{t}sin(t)cos(t)}{ln{10}} + 14te^{t}lg(t)cos^{2}(t) - 11t^{2}e^{t}lg(t)sin(t)cos(t) - 14te^{t}lg(t)sin^{2}(t) - \frac{9te^{t}sin(t)cos(t)}{ln{10}} - t^{2}e^{t}lg(t)cos^{2}(t) + t^{2}e^{t}lg(t)sin^{2}(t) + \frac{6te^{t}cos^{2}(t)}{ln{10}} + \frac{11e^{t}sin(t)cos(t)}{t^{2}ln{10}} - \frac{3e^{t}cos^{2}(t)}{ln{10}} + \frac{3e^{t}sin^{2}(t)}{ln{10}} - \frac{24e^{t}cos^{2}(t)}{tln{10}} + \frac{23e^{t}sin(t)cos(t)}{tln{10}} - \frac{6e^{t}sin(t)cos(t)}{t^{3}ln{10}} + \frac{9e^{t}cos^{2}(t)}{t^{2}ln{10}} + \frac{24e^{t}sin^{2}(t)}{tln{10}} - 3e^{t}lg(t)cos^{2}(t) + 3e^{t}lg(t)sin^{2}(t) - \frac{6te^{t}sin^{2}(t)}{ln{10}} - \frac{9e^{t}sin^{2}(t)}{t^{2}ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 57e^{t}lg(t)sin(t)cos(t) + 4te^{t}lg(t)sin(t)cos(t) + \frac{27e^{t}sin(t)cos(t)}{ln{10}} + 14te^{t}lg(t)cos^{2}(t) - 11t^{2}e^{t}lg(t)sin(t)cos(t) - 14te^{t}lg(t)sin^{2}(t) - \frac{9te^{t}sin(t)cos(t)}{ln{10}} - t^{2}e^{t}lg(t)cos^{2}(t) + t^{2}e^{t}lg(t)sin^{2}(t) + \frac{6te^{t}cos^{2}(t)}{ln{10}} + \frac{11e^{t}sin(t)cos(t)}{t^{2}ln{10}} - \frac{3e^{t}cos^{2}(t)}{ln{10}} + \frac{3e^{t}sin^{2}(t)}{ln{10}} - \frac{24e^{t}cos^{2}(t)}{tln{10}} + \frac{23e^{t}sin(t)cos(t)}{tln{10}} - \frac{6e^{t}sin(t)cos(t)}{t^{3}ln{10}} + \frac{9e^{t}cos^{2}(t)}{t^{2}ln{10}} + \frac{24e^{t}sin^{2}(t)}{tln{10}} - 3e^{t}lg(t)cos^{2}(t) + 3e^{t}lg(t)sin^{2}(t) - \frac{6te^{t}sin^{2}(t)}{ln{10}} - \frac{9e^{t}sin^{2}(t)}{t^{2}ln{10}}\right)}{dt}\\=&57e^{t}lg(t)sin(t)cos(t) + \frac{57e^{t}sin(t)cos(t)}{ln{10}(t)} + 57e^{t}lg(t)cos(t)cos(t) + 57e^{t}lg(t)sin(t)*-sin(t) + 4e^{t}lg(t)sin(t)cos(t) + 4te^{t}lg(t)sin(t)cos(t) + \frac{4te^{t}sin(t)cos(t)}{ln{10}(t)} + 4te^{t}lg(t)cos(t)cos(t) + 4te^{t}lg(t)sin(t)*-sin(t) + \frac{27e^{t}sin(t)cos(t)}{ln{10}} + \frac{27e^{t}*-0sin(t)cos(t)}{ln^{2}{10}} + \frac{27e^{t}cos(t)cos(t)}{ln{10}} + \frac{27e^{t}sin(t)*-sin(t)}{ln{10}} + 14e^{t}lg(t)cos^{2}(t) + 14te^{t}lg(t)cos^{2}(t) + \frac{14te^{t}cos^{2}(t)}{ln{10}(t)} + 14te^{t}lg(t)*-2cos(t)sin(t) - 11*2te^{t}lg(t)sin(t)cos(t) - 11t^{2}e^{t}lg(t)sin(t)cos(t) - \frac{11t^{2}e^{t}sin(t)cos(t)}{ln{10}(t)} - 11t^{2}e^{t}lg(t)cos(t)cos(t) - 11t^{2}e^{t}lg(t)sin(t)*-sin(t) - 14e^{t}lg(t)sin^{2}(t) - 14te^{t}lg(t)sin^{2}(t) - \frac{14te^{t}sin^{2}(t)}{ln{10}(t)} - 14te^{t}lg(t)*2sin(t)cos(t) - \frac{9e^{t}sin(t)cos(t)}{ln{10}} - \frac{9te^{t}sin(t)cos(t)}{ln{10}} - \frac{9te^{t}*-0sin(t)cos(t)}{ln^{2}{10}} - \frac{9te^{t}cos(t)cos(t)}{ln{10}} - \frac{9te^{t}sin(t)*-sin(t)}{ln{10}} - 2te^{t}lg(t)cos^{2}(t) - t^{2}e^{t}lg(t)cos^{2}(t) - \frac{t^{2}e^{t}cos^{2}(t)}{ln{10}(t)} - t^{2}e^{t}lg(t)*-2cos(t)sin(t) + 2te^{t}lg(t)sin^{2}(t) + t^{2}e^{t}lg(t)sin^{2}(t) + \frac{t^{2}e^{t}sin^{2}(t)}{ln{10}(t)} + t^{2}e^{t}lg(t)*2sin(t)cos(t) + \frac{6e^{t}cos^{2}(t)}{ln{10}} + \frac{6te^{t}cos^{2}(t)}{ln{10}} + \frac{6te^{t}*-0cos^{2}(t)}{ln^{2}{10}} + \frac{6te^{t}*-2cos(t)sin(t)}{ln{10}} + \frac{11*-2e^{t}sin(t)cos(t)}{t^{3}ln{10}} + \frac{11e^{t}sin(t)cos(t)}{t^{2}ln{10}} + \frac{11e^{t}*-0sin(t)cos(t)}{t^{2}ln^{2}{10}} + \frac{11e^{t}cos(t)cos(t)}{t^{2}ln{10}} + \frac{11e^{t}sin(t)*-sin(t)}{t^{2}ln{10}} - \frac{3e^{t}cos^{2}(t)}{ln{10}} - \frac{3e^{t}*-0cos^{2}(t)}{ln^{2}{10}} - \frac{3e^{t}*-2cos(t)sin(t)}{ln{10}} + \frac{3e^{t}sin^{2}(t)}{ln{10}} + \frac{3e^{t}*-0sin^{2}(t)}{ln^{2}{10}} + \frac{3e^{t}*2sin(t)cos(t)}{ln{10}} - \frac{24*-e^{t}cos^{2}(t)}{t^{2}ln{10}} - \frac{24e^{t}cos^{2}(t)}{tln{10}} - \frac{24e^{t}*-0cos^{2}(t)}{tln^{2}{10}} - \frac{24e^{t}*-2cos(t)sin(t)}{tln{10}} + \frac{23*-e^{t}sin(t)cos(t)}{t^{2}ln{10}} + \frac{23e^{t}sin(t)cos(t)}{tln{10}} + \frac{23e^{t}*-0sin(t)cos(t)}{tln^{2}{10}} + \frac{23e^{t}cos(t)cos(t)}{tln{10}} + \frac{23e^{t}sin(t)*-sin(t)}{tln{10}} - \frac{6*-3e^{t}sin(t)cos(t)}{t^{4}ln{10}} - \frac{6e^{t}sin(t)cos(t)}{t^{3}ln{10}} - \frac{6e^{t}*-0sin(t)cos(t)}{t^{3}ln^{2}{10}} - \frac{6e^{t}cos(t)cos(t)}{t^{3}ln{10}} - \frac{6e^{t}sin(t)*-sin(t)}{t^{3}ln{10}} + \frac{9*-2e^{t}cos^{2}(t)}{t^{3}ln{10}} + \frac{9e^{t}cos^{2}(t)}{t^{2}ln{10}} + \frac{9e^{t}*-0cos^{2}(t)}{t^{2}ln^{2}{10}} + \frac{9e^{t}*-2cos(t)sin(t)}{t^{2}ln{10}} + \frac{24*-e^{t}sin^{2}(t)}{t^{2}ln{10}} + \frac{24e^{t}sin^{2}(t)}{tln{10}} + \frac{24e^{t}*-0sin^{2}(t)}{tln^{2}{10}} + \frac{24e^{t}*2sin(t)cos(t)}{tln{10}} - 3e^{t}lg(t)cos^{2}(t) - \frac{3e^{t}cos^{2}(t)}{ln{10}(t)} - 3e^{t}lg(t)*-2cos(t)sin(t) + 3e^{t}lg(t)sin^{2}(t) + \frac{3e^{t}sin^{2}(t)}{ln{10}(t)} + 3e^{t}lg(t)*2sin(t)cos(t) - \frac{6e^{t}sin^{2}(t)}{ln{10}} - \frac{6te^{t}sin^{2}(t)}{ln{10}} - \frac{6te^{t}*-0sin^{2}(t)}{ln^{2}{10}} - \frac{6te^{t}*2sin(t)cos(t)}{ln{10}} - \frac{9*-2e^{t}sin^{2}(t)}{t^{3}ln{10}} - \frac{9e^{t}sin^{2}(t)}{t^{2}ln{10}} - \frac{9e^{t}*-0sin^{2}(t)}{t^{2}ln^{2}{10}} - \frac{9e^{t}*2sin(t)cos(t)}{t^{2}ln{10}}\\=&73e^{t}lg(t)sin(t)cos(t) + \frac{176e^{t}sin(t)cos(t)}{tln{10}} + 68e^{t}lg(t)cos^{2}(t) - 68e^{t}lg(t)sin^{2}(t) - 74te^{t}lg(t)sin(t)cos(t) + \frac{34e^{t}sin(t)cos(t)}{ln{10}} + 16te^{t}lg(t)cos^{2}(t) - 7t^{2}e^{t}lg(t)sin(t)cos(t) + \frac{44e^{t}cos^{2}(t)}{ln{10}} - \frac{44e^{t}sin^{2}(t)}{ln{10}} - 12t^{2}e^{t}lg(t)cos^{2}(t) - 16te^{t}lg(t)sin^{2}(t) - \frac{44te^{t}sin(t)cos(t)}{ln{10}} + 12t^{2}e^{t}lg(t)sin^{2}(t) - \frac{4te^{t}cos^{2}(t)}{ln{10}} - \frac{28e^{t}sin(t)cos(t)}{t^{3}ln{10}} + \frac{44e^{t}cos^{2}(t)}{t^{2}ln{10}} - \frac{48e^{t}sin(t)cos(t)}{t^{2}ln{10}} + \frac{18e^{t}sin(t)cos(t)}{t^{4}ln{10}} - \frac{4e^{t}cos^{2}(t)}{tln{10}} - \frac{44e^{t}sin^{2}(t)}{t^{2}ln{10}} - \frac{24e^{t}cos^{2}(t)}{t^{3}ln{10}} + \frac{4e^{t}sin^{2}(t)}{tln{10}} + \frac{4te^{t}sin^{2}(t)}{ln{10}} + \frac{24e^{t}sin^{2}(t)}{t^{3}ln{10}}\\ \end{split}\end{equation} \]