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 y is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ (e^{x} - 6e^{3x})cos(e^{x} + lg(y)) + (e^{4x} - 7e^{2x})sin(e^{x} + lg(y))\ with\ respect\ to\ y：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{x}cos(e^{x} + lg(y)) - 6e^{3x}cos(e^{x} + lg(y)) + e^{4x}sin(e^{x} + lg(y)) - 7e^{2x}sin(e^{x} + lg(y))\right)}{dy}\\=&e^{x}*0cos(e^{x} + lg(y)) + e^{x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)}) - 6e^{3x}*0cos(e^{x} + lg(y)) - 6e^{3x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)}) + e^{4x}*0sin(e^{x} + lg(y)) + e^{4x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)}) - 7e^{2x}*0sin(e^{x} + lg(y)) - 7e^{2x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})\\=& - \frac{e^{x}sin(e^{x} + lg(y))}{yln{10}} + \frac{6e^{3x}sin(e^{x} + lg(y))}{yln{10}} + \frac{e^{4x}cos(e^{x} + lg(y))}{yln{10}} - \frac{7e^{2x}cos(e^{x} + lg(y))}{yln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( - \frac{e^{x}sin(e^{x} + lg(y))}{yln{10}} + \frac{6e^{3x}sin(e^{x} + lg(y))}{yln{10}} + \frac{e^{4x}cos(e^{x} + lg(y))}{yln{10}} - \frac{7e^{2x}cos(e^{x} + lg(y))}{yln{10}}\right)}{dy}\\=& - \frac{-e^{x}sin(e^{x} + lg(y))}{y^{2}ln{10}} - \frac{e^{x}*0sin(e^{x} + lg(y))}{yln{10}} - \frac{e^{x}*-0sin(e^{x} + lg(y))}{yln^{2}{10}} - \frac{e^{x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{yln{10}} + \frac{6*-e^{3x}sin(e^{x} + lg(y))}{y^{2}ln{10}} + \frac{6e^{3x}*0sin(e^{x} + lg(y))}{yln{10}} + \frac{6e^{3x}*-0sin(e^{x} + lg(y))}{yln^{2}{10}} + \frac{6e^{3x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{yln{10}} + \frac{-e^{4x}cos(e^{x} + lg(y))}{y^{2}ln{10}} + \frac{e^{4x}*0cos(e^{x} + lg(y))}{yln{10}} + \frac{e^{4x}*-0cos(e^{x} + lg(y))}{yln^{2}{10}} + \frac{e^{4x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{yln{10}} - \frac{7*-e^{2x}cos(e^{x} + lg(y))}{y^{2}ln{10}} - \frac{7e^{2x}*0cos(e^{x} + lg(y))}{yln{10}} - \frac{7e^{2x}*-0cos(e^{x} + lg(y))}{yln^{2}{10}} - \frac{7e^{2x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{yln{10}}\\=&\frac{e^{x}sin(e^{x} + lg(y))}{y^{2}ln{10}} - \frac{e^{x}cos(e^{x} + lg(y))}{y^{2}ln^{2}{10}} - \frac{6e^{3x}sin(e^{x} + lg(y))}{y^{2}ln{10}} + \frac{6e^{3x}cos(e^{x} + lg(y))}{y^{2}ln^{2}{10}} - \frac{e^{4x}cos(e^{x} + lg(y))}{y^{2}ln{10}} - \frac{e^{4x}sin(e^{x} + lg(y))}{y^{2}ln^{2}{10}} + \frac{7e^{2x}cos(e^{x} + lg(y))}{y^{2}ln{10}} + \frac{7e^{2x}sin(e^{x} + lg(y))}{y^{2}ln^{2}{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{e^{x}sin(e^{x} + lg(y))}{y^{2}ln{10}} - \frac{e^{x}cos(e^{x} + lg(y))}{y^{2}ln^{2}{10}} - \frac{6e^{3x}sin(e^{x} + lg(y))}{y^{2}ln{10}} + \frac{6e^{3x}cos(e^{x} + lg(y))}{y^{2}ln^{2}{10}} - \frac{e^{4x}cos(e^{x} + lg(y))}{y^{2}ln{10}} - \frac{e^{4x}sin(e^{x} + lg(y))}{y^{2}ln^{2}{10}} + \frac{7e^{2x}cos(e^{x} + lg(y))}{y^{2}ln{10}} + \frac{7e^{2x}sin(e^{x} + lg(y))}{y^{2}ln^{2}{10}}\right)}{dy}\\=&\frac{-2e^{x}sin(e^{x} + lg(y))}{y^{3}ln{10}} + \frac{e^{x}*0sin(e^{x} + lg(y))}{y^{2}ln{10}} + \frac{e^{x}*-0sin(e^{x} + lg(y))}{y^{2}ln^{2}{10}} + \frac{e^{x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{2}ln{10}} - \frac{-2e^{x}cos(e^{x} + lg(y))}{y^{3}ln^{2}{10}} - \frac{e^{x}*0cos(e^{x} + lg(y))}{y^{2}ln^{2}{10}} - \frac{e^{x}*-2*0cos(e^{x} + lg(y))}{y^{2}ln^{3}{10}} - \frac{e^{x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{2}ln^{2}{10}} - \frac{6*-2e^{3x}sin(e^{x} + lg(y))}{y^{3}ln{10}} - \frac{6e^{3x}*0sin(e^{x} + lg(y))}{y^{2}ln{10}} - \frac{6e^{3x}*-0sin(e^{x} + lg(y))}{y^{2}ln^{2}{10}} - \frac{6e^{3x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{2}ln{10}} + \frac{6*-2e^{3x}cos(e^{x} + lg(y))}{y^{3}ln^{2}{10}} + \frac{6e^{3x}*0cos(e^{x} + lg(y))}{y^{2}ln^{2}{10}} + \frac{6e^{3x}*-2*0cos(e^{x} + lg(y))}{y^{2}ln^{3}{10}} + \frac{6e^{3x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{2}ln^{2}{10}} - \frac{-2e^{4x}cos(e^{x} + lg(y))}{y^{3}ln{10}} - \frac{e^{4x}*0cos(e^{x} + lg(y))}{y^{2}ln{10}} - \frac{e^{4x}*-0cos(e^{x} + lg(y))}{y^{2}ln^{2}{10}} - \frac{e^{4x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{2}ln{10}} - \frac{-2e^{4x}sin(e^{x} + lg(y))}{y^{3}ln^{2}{10}} - \frac{e^{4x}*0sin(e^{x} + lg(y))}{y^{2}ln^{2}{10}} - \frac{e^{4x}*-2*0sin(e^{x} + lg(y))}{y^{2}ln^{3}{10}} - \frac{e^{4x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{2}ln^{2}{10}} + \frac{7*-2e^{2x}cos(e^{x} + lg(y))}{y^{3}ln{10}} + \frac{7e^{2x}*0cos(e^{x} + lg(y))}{y^{2}ln{10}} + \frac{7e^{2x}*-0cos(e^{x} + lg(y))}{y^{2}ln^{2}{10}} + \frac{7e^{2x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{2}ln{10}} + \frac{7*-2e^{2x}sin(e^{x} + lg(y))}{y^{3}ln^{2}{10}} + \frac{7e^{2x}*0sin(e^{x} + lg(y))}{y^{2}ln^{2}{10}} + \frac{7e^{2x}*-2*0sin(e^{x} + lg(y))}{y^{2}ln^{3}{10}} + \frac{7e^{2x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{2}ln^{2}{10}}\\=& - \frac{2e^{x}sin(e^{x} + lg(y))}{y^{3}ln{10}} + \frac{3e^{x}cos(e^{x} + lg(y))}{y^{3}ln^{2}{10}} + \frac{e^{x}sin(e^{x} + lg(y))}{y^{3}ln^{3}{10}} + \frac{12e^{3x}sin(e^{x} + lg(y))}{y^{3}ln{10}} - \frac{18e^{3x}cos(e^{x} + lg(y))}{y^{3}ln^{2}{10}} - \frac{6e^{3x}sin(e^{x} + lg(y))}{y^{3}ln^{3}{10}} + \frac{2e^{4x}cos(e^{x} + lg(y))}{y^{3}ln{10}} + \frac{3e^{4x}sin(e^{x} + lg(y))}{y^{3}ln^{2}{10}} - \frac{e^{4x}cos(e^{x} + lg(y))}{y^{3}ln^{3}{10}} - \frac{14e^{2x}cos(e^{x} + lg(y))}{y^{3}ln{10}} - \frac{21e^{2x}sin(e^{x} + lg(y))}{y^{3}ln^{2}{10}} + \frac{7e^{2x}cos(e^{x} + lg(y))}{y^{3}ln^{3}{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - \frac{2e^{x}sin(e^{x} + lg(y))}{y^{3}ln{10}} + \frac{3e^{x}cos(e^{x} + lg(y))}{y^{3}ln^{2}{10}} + \frac{e^{x}sin(e^{x} + lg(y))}{y^{3}ln^{3}{10}} + \frac{12e^{3x}sin(e^{x} + lg(y))}{y^{3}ln{10}} - \frac{18e^{3x}cos(e^{x} + lg(y))}{y^{3}ln^{2}{10}} - \frac{6e^{3x}sin(e^{x} + lg(y))}{y^{3}ln^{3}{10}} + \frac{2e^{4x}cos(e^{x} + lg(y))}{y^{3}ln{10}} + \frac{3e^{4x}sin(e^{x} + lg(y))}{y^{3}ln^{2}{10}} - \frac{e^{4x}cos(e^{x} + lg(y))}{y^{3}ln^{3}{10}} - \frac{14e^{2x}cos(e^{x} + lg(y))}{y^{3}ln{10}} - \frac{21e^{2x}sin(e^{x} + lg(y))}{y^{3}ln^{2}{10}} + \frac{7e^{2x}cos(e^{x} + lg(y))}{y^{3}ln^{3}{10}}\right)}{dy}\\=& - \frac{2*-3e^{x}sin(e^{x} + lg(y))}{y^{4}ln{10}} - \frac{2e^{x}*0sin(e^{x} + lg(y))}{y^{3}ln{10}} - \frac{2e^{x}*-0sin(e^{x} + lg(y))}{y^{3}ln^{2}{10}} - \frac{2e^{x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln{10}} + \frac{3*-3e^{x}cos(e^{x} + lg(y))}{y^{4}ln^{2}{10}} + \frac{3e^{x}*0cos(e^{x} + lg(y))}{y^{3}ln^{2}{10}} + \frac{3e^{x}*-2*0cos(e^{x} + lg(y))}{y^{3}ln^{3}{10}} + \frac{3e^{x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln^{2}{10}} + \frac{-3e^{x}sin(e^{x} + lg(y))}{y^{4}ln^{3}{10}} + \frac{e^{x}*0sin(e^{x} + lg(y))}{y^{3}ln^{3}{10}} + \frac{e^{x}*-3*0sin(e^{x} + lg(y))}{y^{3}ln^{4}{10}} + \frac{e^{x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln^{3}{10}} + \frac{12*-3e^{3x}sin(e^{x} + lg(y))}{y^{4}ln{10}} + \frac{12e^{3x}*0sin(e^{x} + lg(y))}{y^{3}ln{10}} + \frac{12e^{3x}*-0sin(e^{x} + lg(y))}{y^{3}ln^{2}{10}} + \frac{12e^{3x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln{10}} - \frac{18*-3e^{3x}cos(e^{x} + lg(y))}{y^{4}ln^{2}{10}} - \frac{18e^{3x}*0cos(e^{x} + lg(y))}{y^{3}ln^{2}{10}} - \frac{18e^{3x}*-2*0cos(e^{x} + lg(y))}{y^{3}ln^{3}{10}} - \frac{18e^{3x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln^{2}{10}} - \frac{6*-3e^{3x}sin(e^{x} + lg(y))}{y^{4}ln^{3}{10}} - \frac{6e^{3x}*0sin(e^{x} + lg(y))}{y^{3}ln^{3}{10}} - \frac{6e^{3x}*-3*0sin(e^{x} + lg(y))}{y^{3}ln^{4}{10}} - \frac{6e^{3x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln^{3}{10}} + \frac{2*-3e^{4x}cos(e^{x} + lg(y))}{y^{4}ln{10}} + \frac{2e^{4x}*0cos(e^{x} + lg(y))}{y^{3}ln{10}} + \frac{2e^{4x}*-0cos(e^{x} + lg(y))}{y^{3}ln^{2}{10}} + \frac{2e^{4x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln{10}} + \frac{3*-3e^{4x}sin(e^{x} + lg(y))}{y^{4}ln^{2}{10}} + \frac{3e^{4x}*0sin(e^{x} + lg(y))}{y^{3}ln^{2}{10}} + \frac{3e^{4x}*-2*0sin(e^{x} + lg(y))}{y^{3}ln^{3}{10}} + \frac{3e^{4x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln^{2}{10}} - \frac{-3e^{4x}cos(e^{x} + lg(y))}{y^{4}ln^{3}{10}} - \frac{e^{4x}*0cos(e^{x} + lg(y))}{y^{3}ln^{3}{10}} - \frac{e^{4x}*-3*0cos(e^{x} + lg(y))}{y^{3}ln^{4}{10}} - \frac{e^{4x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln^{3}{10}} - \frac{14*-3e^{2x}cos(e^{x} + lg(y))}{y^{4}ln{10}} - \frac{14e^{2x}*0cos(e^{x} + lg(y))}{y^{3}ln{10}} - \frac{14e^{2x}*-0cos(e^{x} + lg(y))}{y^{3}ln^{2}{10}} - \frac{14e^{2x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln{10}} - \frac{21*-3e^{2x}sin(e^{x} + lg(y))}{y^{4}ln^{2}{10}} - \frac{21e^{2x}*0sin(e^{x} + lg(y))}{y^{3}ln^{2}{10}} - \frac{21e^{2x}*-2*0sin(e^{x} + lg(y))}{y^{3}ln^{3}{10}} - \frac{21e^{2x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln^{2}{10}} + \frac{7*-3e^{2x}cos(e^{x} + lg(y))}{y^{4}ln^{3}{10}} + \frac{7e^{2x}*0cos(e^{x} + lg(y))}{y^{3}ln^{3}{10}} + \frac{7e^{2x}*-3*0cos(e^{x} + lg(y))}{y^{3}ln^{4}{10}} + \frac{7e^{2x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln^{3}{10}}\\=&\frac{6e^{x}sin(e^{x} + lg(y))}{y^{4}ln{10}} - \frac{11e^{x}cos(e^{x} + lg(y))}{y^{4}ln^{2}{10}} - \frac{6e^{x}sin(e^{x} + lg(y))}{y^{4}ln^{3}{10}} + \frac{e^{x}cos(e^{x} + lg(y))}{y^{4}ln^{4}{10}} - \frac{36e^{3x}sin(e^{x} + lg(y))}{y^{4}ln{10}} + \frac{66e^{3x}cos(e^{x} + lg(y))}{y^{4}ln^{2}{10}} + \frac{36e^{3x}sin(e^{x} + lg(y))}{y^{4}ln^{3}{10}} - \frac{6e^{3x}cos(e^{x} + lg(y))}{y^{4}ln^{4}{10}} - \frac{6e^{4x}cos(e^{x} + lg(y))}{y^{4}ln{10}} - \frac{11e^{4x}sin(e^{x} + lg(y))}{y^{4}ln^{2}{10}} + \frac{6e^{4x}cos(e^{x} + lg(y))}{y^{4}ln^{3}{10}} + \frac{e^{4x}sin(e^{x} + lg(y))}{y^{4}ln^{4}{10}} + \frac{42e^{2x}cos(e^{x} + lg(y))}{y^{4}ln{10}} + \frac{77e^{2x}sin(e^{x} + lg(y))}{y^{4}ln^{2}{10}} - \frac{42e^{2x}cos(e^{x} + lg(y))}{y^{4}ln^{3}{10}} - \frac{7e^{2x}sin(e^{x} + lg(y))}{y^{4}ln^{4}{10}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ (e^{x} - 6e^{3x})cos(e^{x} + lg(y)) + (e^{4x} - 7e^{2x})sin(e^{x} + lg(y))\ with\ respect\ to\ y：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{x}cos(e^{x} + lg(y)) - 6e^{3x}cos(e^{x} + lg(y)) + e^{4x}sin(e^{x} + lg(y)) - 7e^{2x}sin(e^{x} + lg(y))\right)}{dy}\\=&e^{x}*0cos(e^{x} + lg(y)) + e^{x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)}) - 6e^{3x}*0cos(e^{x} + lg(y)) - 6e^{3x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)}) + e^{4x}*0sin(e^{x} + lg(y)) + e^{4x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)}) - 7e^{2x}*0sin(e^{x} + lg(y)) - 7e^{2x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})\\=& - \frac{e^{x}sin(e^{x} + lg(y))}{yln{10}} + \frac{6e^{3x}sin(e^{x} + lg(y))}{yln{10}} + \frac{e^{4x}cos(e^{x} + lg(y))}{yln{10}} - \frac{7e^{2x}cos(e^{x} + lg(y))}{yln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( - \frac{e^{x}sin(e^{x} + lg(y))}{yln{10}} + \frac{6e^{3x}sin(e^{x} + lg(y))}{yln{10}} + \frac{e^{4x}cos(e^{x} + lg(y))}{yln{10}} - \frac{7e^{2x}cos(e^{x} + lg(y))}{yln{10}}\right)}{dy}\\=& - \frac{-e^{x}sin(e^{x} + lg(y))}{y^{2}ln{10}} - \frac{e^{x}*0sin(e^{x} + lg(y))}{yln{10}} - \frac{e^{x}*-0sin(e^{x} + lg(y))}{yln^{2}{10}} - \frac{e^{x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{yln{10}} + \frac{6*-e^{3x}sin(e^{x} + lg(y))}{y^{2}ln{10}} + \frac{6e^{3x}*0sin(e^{x} + lg(y))}{yln{10}} + \frac{6e^{3x}*-0sin(e^{x} + lg(y))}{yln^{2}{10}} + \frac{6e^{3x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{yln{10}} + \frac{-e^{4x}cos(e^{x} + lg(y))}{y^{2}ln{10}} + \frac{e^{4x}*0cos(e^{x} + lg(y))}{yln{10}} + \frac{e^{4x}*-0cos(e^{x} + lg(y))}{yln^{2}{10}} + \frac{e^{4x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{yln{10}} - \frac{7*-e^{2x}cos(e^{x} + lg(y))}{y^{2}ln{10}} - \frac{7e^{2x}*0cos(e^{x} + lg(y))}{yln{10}} - \frac{7e^{2x}*-0cos(e^{x} + lg(y))}{yln^{2}{10}} - \frac{7e^{2x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{yln{10}}\\=&\frac{e^{x}sin(e^{x} + lg(y))}{y^{2}ln{10}} - \frac{e^{x}cos(e^{x} + lg(y))}{y^{2}ln^{2}{10}} - \frac{6e^{3x}sin(e^{x} + lg(y))}{y^{2}ln{10}} + \frac{6e^{3x}cos(e^{x} + lg(y))}{y^{2}ln^{2}{10}} - \frac{e^{4x}cos(e^{x} + lg(y))}{y^{2}ln{10}} - \frac{e^{4x}sin(e^{x} + lg(y))}{y^{2}ln^{2}{10}} + \frac{7e^{2x}cos(e^{x} + lg(y))}{y^{2}ln{10}} + \frac{7e^{2x}sin(e^{x} + lg(y))}{y^{2}ln^{2}{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{e^{x}sin(e^{x} + lg(y))}{y^{2}ln{10}} - \frac{e^{x}cos(e^{x} + lg(y))}{y^{2}ln^{2}{10}} - \frac{6e^{3x}sin(e^{x} + lg(y))}{y^{2}ln{10}} + \frac{6e^{3x}cos(e^{x} + lg(y))}{y^{2}ln^{2}{10}} - \frac{e^{4x}cos(e^{x} + lg(y))}{y^{2}ln{10}} - \frac{e^{4x}sin(e^{x} + lg(y))}{y^{2}ln^{2}{10}} + \frac{7e^{2x}cos(e^{x} + lg(y))}{y^{2}ln{10}} + \frac{7e^{2x}sin(e^{x} + lg(y))}{y^{2}ln^{2}{10}}\right)}{dy}\\=&\frac{-2e^{x}sin(e^{x} + lg(y))}{y^{3}ln{10}} + \frac{e^{x}*0sin(e^{x} + lg(y))}{y^{2}ln{10}} + \frac{e^{x}*-0sin(e^{x} + lg(y))}{y^{2}ln^{2}{10}} + \frac{e^{x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{2}ln{10}} - \frac{-2e^{x}cos(e^{x} + lg(y))}{y^{3}ln^{2}{10}} - \frac{e^{x}*0cos(e^{x} + lg(y))}{y^{2}ln^{2}{10}} - \frac{e^{x}*-2*0cos(e^{x} + lg(y))}{y^{2}ln^{3}{10}} - \frac{e^{x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{2}ln^{2}{10}} - \frac{6*-2e^{3x}sin(e^{x} + lg(y))}{y^{3}ln{10}} - \frac{6e^{3x}*0sin(e^{x} + lg(y))}{y^{2}ln{10}} - \frac{6e^{3x}*-0sin(e^{x} + lg(y))}{y^{2}ln^{2}{10}} - \frac{6e^{3x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{2}ln{10}} + \frac{6*-2e^{3x}cos(e^{x} + lg(y))}{y^{3}ln^{2}{10}} + \frac{6e^{3x}*0cos(e^{x} + lg(y))}{y^{2}ln^{2}{10}} + \frac{6e^{3x}*-2*0cos(e^{x} + lg(y))}{y^{2}ln^{3}{10}} + \frac{6e^{3x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{2}ln^{2}{10}} - \frac{-2e^{4x}cos(e^{x} + lg(y))}{y^{3}ln{10}} - \frac{e^{4x}*0cos(e^{x} + lg(y))}{y^{2}ln{10}} - \frac{e^{4x}*-0cos(e^{x} + lg(y))}{y^{2}ln^{2}{10}} - \frac{e^{4x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{2}ln{10}} - \frac{-2e^{4x}sin(e^{x} + lg(y))}{y^{3}ln^{2}{10}} - \frac{e^{4x}*0sin(e^{x} + lg(y))}{y^{2}ln^{2}{10}} - \frac{e^{4x}*-2*0sin(e^{x} + lg(y))}{y^{2}ln^{3}{10}} - \frac{e^{4x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{2}ln^{2}{10}} + \frac{7*-2e^{2x}cos(e^{x} + lg(y))}{y^{3}ln{10}} + \frac{7e^{2x}*0cos(e^{x} + lg(y))}{y^{2}ln{10}} + \frac{7e^{2x}*-0cos(e^{x} + lg(y))}{y^{2}ln^{2}{10}} + \frac{7e^{2x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{2}ln{10}} + \frac{7*-2e^{2x}sin(e^{x} + lg(y))}{y^{3}ln^{2}{10}} + \frac{7e^{2x}*0sin(e^{x} + lg(y))}{y^{2}ln^{2}{10}} + \frac{7e^{2x}*-2*0sin(e^{x} + lg(y))}{y^{2}ln^{3}{10}} + \frac{7e^{2x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{2}ln^{2}{10}}\\=& - \frac{2e^{x}sin(e^{x} + lg(y))}{y^{3}ln{10}} + \frac{3e^{x}cos(e^{x} + lg(y))}{y^{3}ln^{2}{10}} + \frac{e^{x}sin(e^{x} + lg(y))}{y^{3}ln^{3}{10}} + \frac{12e^{3x}sin(e^{x} + lg(y))}{y^{3}ln{10}} - \frac{18e^{3x}cos(e^{x} + lg(y))}{y^{3}ln^{2}{10}} - \frac{6e^{3x}sin(e^{x} + lg(y))}{y^{3}ln^{3}{10}} + \frac{2e^{4x}cos(e^{x} + lg(y))}{y^{3}ln{10}} + \frac{3e^{4x}sin(e^{x} + lg(y))}{y^{3}ln^{2}{10}} - \frac{e^{4x}cos(e^{x} + lg(y))}{y^{3}ln^{3}{10}} - \frac{14e^{2x}cos(e^{x} + lg(y))}{y^{3}ln{10}} - \frac{21e^{2x}sin(e^{x} + lg(y))}{y^{3}ln^{2}{10}} + \frac{7e^{2x}cos(e^{x} + lg(y))}{y^{3}ln^{3}{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - \frac{2e^{x}sin(e^{x} + lg(y))}{y^{3}ln{10}} + \frac{3e^{x}cos(e^{x} + lg(y))}{y^{3}ln^{2}{10}} + \frac{e^{x}sin(e^{x} + lg(y))}{y^{3}ln^{3}{10}} + \frac{12e^{3x}sin(e^{x} + lg(y))}{y^{3}ln{10}} - \frac{18e^{3x}cos(e^{x} + lg(y))}{y^{3}ln^{2}{10}} - \frac{6e^{3x}sin(e^{x} + lg(y))}{y^{3}ln^{3}{10}} + \frac{2e^{4x}cos(e^{x} + lg(y))}{y^{3}ln{10}} + \frac{3e^{4x}sin(e^{x} + lg(y))}{y^{3}ln^{2}{10}} - \frac{e^{4x}cos(e^{x} + lg(y))}{y^{3}ln^{3}{10}} - \frac{14e^{2x}cos(e^{x} + lg(y))}{y^{3}ln{10}} - \frac{21e^{2x}sin(e^{x} + lg(y))}{y^{3}ln^{2}{10}} + \frac{7e^{2x}cos(e^{x} + lg(y))}{y^{3}ln^{3}{10}}\right)}{dy}\\=& - \frac{2*-3e^{x}sin(e^{x} + lg(y))}{y^{4}ln{10}} - \frac{2e^{x}*0sin(e^{x} + lg(y))}{y^{3}ln{10}} - \frac{2e^{x}*-0sin(e^{x} + lg(y))}{y^{3}ln^{2}{10}} - \frac{2e^{x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln{10}} + \frac{3*-3e^{x}cos(e^{x} + lg(y))}{y^{4}ln^{2}{10}} + \frac{3e^{x}*0cos(e^{x} + lg(y))}{y^{3}ln^{2}{10}} + \frac{3e^{x}*-2*0cos(e^{x} + lg(y))}{y^{3}ln^{3}{10}} + \frac{3e^{x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln^{2}{10}} + \frac{-3e^{x}sin(e^{x} + lg(y))}{y^{4}ln^{3}{10}} + \frac{e^{x}*0sin(e^{x} + lg(y))}{y^{3}ln^{3}{10}} + \frac{e^{x}*-3*0sin(e^{x} + lg(y))}{y^{3}ln^{4}{10}} + \frac{e^{x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln^{3}{10}} + \frac{12*-3e^{3x}sin(e^{x} + lg(y))}{y^{4}ln{10}} + \frac{12e^{3x}*0sin(e^{x} + lg(y))}{y^{3}ln{10}} + \frac{12e^{3x}*-0sin(e^{x} + lg(y))}{y^{3}ln^{2}{10}} + \frac{12e^{3x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln{10}} - \frac{18*-3e^{3x}cos(e^{x} + lg(y))}{y^{4}ln^{2}{10}} - \frac{18e^{3x}*0cos(e^{x} + lg(y))}{y^{3}ln^{2}{10}} - \frac{18e^{3x}*-2*0cos(e^{x} + lg(y))}{y^{3}ln^{3}{10}} - \frac{18e^{3x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln^{2}{10}} - \frac{6*-3e^{3x}sin(e^{x} + lg(y))}{y^{4}ln^{3}{10}} - \frac{6e^{3x}*0sin(e^{x} + lg(y))}{y^{3}ln^{3}{10}} - \frac{6e^{3x}*-3*0sin(e^{x} + lg(y))}{y^{3}ln^{4}{10}} - \frac{6e^{3x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln^{3}{10}} + \frac{2*-3e^{4x}cos(e^{x} + lg(y))}{y^{4}ln{10}} + \frac{2e^{4x}*0cos(e^{x} + lg(y))}{y^{3}ln{10}} + \frac{2e^{4x}*-0cos(e^{x} + lg(y))}{y^{3}ln^{2}{10}} + \frac{2e^{4x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln{10}} + \frac{3*-3e^{4x}sin(e^{x} + lg(y))}{y^{4}ln^{2}{10}} + \frac{3e^{4x}*0sin(e^{x} + lg(y))}{y^{3}ln^{2}{10}} + \frac{3e^{4x}*-2*0sin(e^{x} + lg(y))}{y^{3}ln^{3}{10}} + \frac{3e^{4x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln^{2}{10}} - \frac{-3e^{4x}cos(e^{x} + lg(y))}{y^{4}ln^{3}{10}} - \frac{e^{4x}*0cos(e^{x} + lg(y))}{y^{3}ln^{3}{10}} - \frac{e^{4x}*-3*0cos(e^{x} + lg(y))}{y^{3}ln^{4}{10}} - \frac{e^{4x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln^{3}{10}} - \frac{14*-3e^{2x}cos(e^{x} + lg(y))}{y^{4}ln{10}} - \frac{14e^{2x}*0cos(e^{x} + lg(y))}{y^{3}ln{10}} - \frac{14e^{2x}*-0cos(e^{x} + lg(y))}{y^{3}ln^{2}{10}} - \frac{14e^{2x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln{10}} - \frac{21*-3e^{2x}sin(e^{x} + lg(y))}{y^{4}ln^{2}{10}} - \frac{21e^{2x}*0sin(e^{x} + lg(y))}{y^{3}ln^{2}{10}} - \frac{21e^{2x}*-2*0sin(e^{x} + lg(y))}{y^{3}ln^{3}{10}} - \frac{21e^{2x}cos(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln^{2}{10}} + \frac{7*-3e^{2x}cos(e^{x} + lg(y))}{y^{4}ln^{3}{10}} + \frac{7e^{2x}*0cos(e^{x} + lg(y))}{y^{3}ln^{3}{10}} + \frac{7e^{2x}*-3*0cos(e^{x} + lg(y))}{y^{3}ln^{4}{10}} + \frac{7e^{2x}*-sin(e^{x} + lg(y))(e^{x}*0 + \frac{1}{ln{10}(y)})}{y^{3}ln^{3}{10}}\\=&\frac{6e^{x}sin(e^{x} + lg(y))}{y^{4}ln{10}} - \frac{11e^{x}cos(e^{x} + lg(y))}{y^{4}ln^{2}{10}} - \frac{6e^{x}sin(e^{x} + lg(y))}{y^{4}ln^{3}{10}} + \frac{e^{x}cos(e^{x} + lg(y))}{y^{4}ln^{4}{10}} - \frac{36e^{3x}sin(e^{x} + lg(y))}{y^{4}ln{10}} + \frac{66e^{3x}cos(e^{x} + lg(y))}{y^{4}ln^{2}{10}} + \frac{36e^{3x}sin(e^{x} + lg(y))}{y^{4}ln^{3}{10}} - \frac{6e^{3x}cos(e^{x} + lg(y))}{y^{4}ln^{4}{10}} - \frac{6e^{4x}cos(e^{x} + lg(y))}{y^{4}ln{10}} - \frac{11e^{4x}sin(e^{x} + lg(y))}{y^{4}ln^{2}{10}} + \frac{6e^{4x}cos(e^{x} + lg(y))}{y^{4}ln^{3}{10}} + \frac{e^{4x}sin(e^{x} + lg(y))}{y^{4}ln^{4}{10}} + \frac{42e^{2x}cos(e^{x} + lg(y))}{y^{4}ln{10}} + \frac{77e^{2x}sin(e^{x} + lg(y))}{y^{4}ln^{2}{10}} - \frac{42e^{2x}cos(e^{x} + lg(y))}{y^{4}ln^{3}{10}} - \frac{7e^{2x}sin(e^{x} + lg(y))}{y^{4}ln^{4}{10}}\\ \end{split}\end{equation} \]