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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ e^{x(axx + (4a + 1)x + 4a + 3)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}\right)}{dx}\\=&e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3)\\=&3ax^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 8axe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 4ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 3e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 3ax^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 8axe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 4ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 3e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}\right)}{dx}\\=&3a*2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 3ax^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 8ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 8axe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 2e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 4ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 3e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3)\\=&70axe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 9a^{2}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 48a^{2}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 12ax^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 88a^{2}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 50ax^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 32ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 64a^{2}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 11e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 4x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 12xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 16a^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 70axe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 9a^{2}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 48a^{2}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 12ax^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 88a^{2}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 50ax^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 32ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 64a^{2}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 11e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 4x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 12xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 16a^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}\right)}{dx}\\=&70ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 70axe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 9a^{2}*4x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 9a^{2}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 48a^{2}*3x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 48a^{2}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 12a*3x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 12ax^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 88a^{2}*2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 88a^{2}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 50a*2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 50ax^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 32ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 64a^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 64a^{2}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 11e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 4*2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 4x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 12e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 12xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 16a^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3)\\=&210ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1014a^{2}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1392a^{2}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 471ax^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 936a^{2}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 510axe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 27a^{3}x^{6}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 216a^{3}x^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 54a^{2}x^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 684a^{3}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 369a^{2}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1088a^{3}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 36ax^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 204ax^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 912a^{3}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 240a^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 384a^{3}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 66xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 45e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 8x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 36x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 64a^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 210ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1014a^{2}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1392a^{2}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 471ax^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 936a^{2}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 510axe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 27a^{3}x^{6}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 216a^{3}x^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 54a^{2}x^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 684a^{3}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 369a^{2}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1088a^{3}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 36ax^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 204ax^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 912a^{3}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 240a^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 384a^{3}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 66xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 45e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 8x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 36x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 64a^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}\right)}{dx}\\=&210ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 1014a^{2}*3x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1014a^{2}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 1392a^{2}*2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1392a^{2}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 471a*2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 471ax^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 936a^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 936a^{2}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 510ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 510axe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 27a^{3}*6x^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 27a^{3}x^{6}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 216a^{3}*5x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 216a^{3}x^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 54a^{2}*5x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 54a^{2}x^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 684a^{3}*4x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 684a^{3}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 369a^{2}*4x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 369a^{2}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 1088a^{3}*3x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1088a^{3}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 36a*4x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 36ax^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 204a*3x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 204ax^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 912a^{3}*2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 912a^{3}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 240a^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 384a^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 384a^{3}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 66e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 66xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 45e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 8*3x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 8x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 36*2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 36x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 64a^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3)\\=&15684a^{2}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 9792a^{2}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 3516axe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 2496a^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1320ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 8388a^{3}x^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 19072a^{3}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 6594a^{2}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 25824a^{3}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 13416a^{2}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 20544a^{3}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 2216ax^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 3852ax^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 8768a^{3}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 81a^{4}x^{8}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 864a^{4}x^{7}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 216a^{3}x^{7}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 3888a^{4}x^{6}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 2052a^{3}x^{6}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 9600a^{4}x^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 216a^{2}x^{6}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1800a^{2}x^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 14176a^{4}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 12800a^{4}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 96ax^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 688ax^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 6912a^{4}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1536a^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 2048a^{4}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 201e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 264x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 360xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 16x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 96x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 256a^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ e^{x(axx + (4a + 1)x + 4a + 3)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}\right)}{dx}\\=&e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3)\\=&3ax^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 8axe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 4ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 3e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 3ax^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 8axe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 4ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 3e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}\right)}{dx}\\=&3a*2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 3ax^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 8ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 8axe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 2e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 4ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 3e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3)\\=&70axe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 9a^{2}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 48a^{2}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 12ax^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 88a^{2}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 50ax^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 32ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 64a^{2}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 11e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 4x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 12xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 16a^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 70axe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 9a^{2}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 48a^{2}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 12ax^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 88a^{2}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 50ax^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 32ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 64a^{2}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 11e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 4x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 12xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 16a^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}\right)}{dx}\\=&70ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 70axe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 9a^{2}*4x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 9a^{2}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 48a^{2}*3x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 48a^{2}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 12a*3x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 12ax^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 88a^{2}*2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 88a^{2}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 50a*2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 50ax^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 32ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 64a^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 64a^{2}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 11e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 4*2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 4x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 12e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 12xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 16a^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3)\\=&210ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1014a^{2}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1392a^{2}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 471ax^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 936a^{2}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 510axe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 27a^{3}x^{6}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 216a^{3}x^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 54a^{2}x^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 684a^{3}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 369a^{2}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1088a^{3}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 36ax^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 204ax^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 912a^{3}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 240a^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 384a^{3}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 66xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 45e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 8x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 36x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 64a^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 210ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1014a^{2}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1392a^{2}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 471ax^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 936a^{2}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 510axe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 27a^{3}x^{6}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 216a^{3}x^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 54a^{2}x^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 684a^{3}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 369a^{2}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1088a^{3}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 36ax^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 204ax^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 912a^{3}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 240a^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 384a^{3}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 66xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 45e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 8x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 36x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 64a^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}\right)}{dx}\\=&210ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 1014a^{2}*3x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1014a^{2}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 1392a^{2}*2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1392a^{2}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 471a*2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 471ax^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 936a^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 936a^{2}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 510ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 510axe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 27a^{3}*6x^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 27a^{3}x^{6}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 216a^{3}*5x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 216a^{3}x^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 54a^{2}*5x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 54a^{2}x^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 684a^{3}*4x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 684a^{3}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 369a^{2}*4x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 369a^{2}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 1088a^{3}*3x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1088a^{3}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 36a*4x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 36ax^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 204a*3x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 204ax^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 912a^{3}*2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 912a^{3}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 240a^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 384a^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 384a^{3}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 66e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 66xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 45e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 8*3x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 8x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 36*2xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 36x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3) + 64a^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}(a*3x^{2} + 4a*2x + 2x + 4a + 3)\\=&15684a^{2}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 9792a^{2}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 3516axe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 2496a^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1320ae^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 8388a^{3}x^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 19072a^{3}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 6594a^{2}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 25824a^{3}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 13416a^{2}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 20544a^{3}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 2216ax^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 3852ax^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 8768a^{3}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 81a^{4}x^{8}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 864a^{4}x^{7}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 216a^{3}x^{7}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 3888a^{4}x^{6}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 2052a^{3}x^{6}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 9600a^{4}x^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 216a^{2}x^{6}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1800a^{2}x^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 14176a^{4}x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 12800a^{4}x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 96ax^{5}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 688ax^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 6912a^{4}x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 1536a^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 2048a^{4}xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 201e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 264x^{2}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 360xe^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 16x^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 96x^{3}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x} + 256a^{4}e^{ax^{3} + 4ax^{2} + x^{2} + 4ax + 3x}\\ \end{split}\end{equation} \]