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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{(axx + bx + c)}{(dxx + xe + f)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{ax^{2}}{(dx^{2} + xe + f)} + \frac{bx}{(dx^{2} + xe + f)} + \frac{c}{(dx^{2} + xe + f)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{ax^{2}}{(dx^{2} + xe + f)} + \frac{bx}{(dx^{2} + xe + f)} + \frac{c}{(dx^{2} + xe + f)}\right)}{dx}\\=&(\frac{-(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{2}})ax^{2} + \frac{a*2x}{(dx^{2} + xe + f)} + (\frac{-(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{2}})bx + \frac{b}{(dx^{2} + xe + f)} + (\frac{-(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{2}})c + 0\\=&\frac{-2adx^{3}}{(dx^{2} + xe + f)^{2}} - \frac{ax^{2}e}{(dx^{2} + xe + f)^{2}} + \frac{2ax}{(dx^{2} + xe + f)} - \frac{2bdx^{2}}{(dx^{2} + xe + f)^{2}} - \frac{bxe}{(dx^{2} + xe + f)^{2}} + \frac{b}{(dx^{2} + xe + f)} - \frac{2cdx}{(dx^{2} + xe + f)^{2}} - \frac{ce}{(dx^{2} + xe + f)^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2adx^{3}}{(dx^{2} + xe + f)^{2}} - \frac{ax^{2}e}{(dx^{2} + xe + f)^{2}} + \frac{2ax}{(dx^{2} + xe + f)} - \frac{2bdx^{2}}{(dx^{2} + xe + f)^{2}} - \frac{bxe}{(dx^{2} + xe + f)^{2}} + \frac{b}{(dx^{2} + xe + f)} - \frac{2cdx}{(dx^{2} + xe + f)^{2}} - \frac{ce}{(dx^{2} + xe + f)^{2}}\right)}{dx}\\=&-2(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})adx^{3} - \frac{2ad*3x^{2}}{(dx^{2} + xe + f)^{2}} - (\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})ax^{2}e - \frac{a*2xe}{(dx^{2} + xe + f)^{2}} - \frac{ax^{2}*0}{(dx^{2} + xe + f)^{2}} + 2(\frac{-(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{2}})ax + \frac{2a}{(dx^{2} + xe + f)} - 2(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})bdx^{2} - \frac{2bd*2x}{(dx^{2} + xe + f)^{2}} - (\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})bxe - \frac{be}{(dx^{2} + xe + f)^{2}} - \frac{bx*0}{(dx^{2} + xe + f)^{2}} + (\frac{-(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{2}})b + 0 - 2(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})cdx - \frac{2cd}{(dx^{2} + xe + f)^{2}} - (\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})ce - \frac{c*0}{(dx^{2} + xe + f)^{2}}\\=&\frac{8adx^{3}e}{(dx^{2} + xe + f)^{3}} + \frac{8ad^{2}x^{4}}{(dx^{2} + xe + f)^{3}} - \frac{10adx^{2}}{(dx^{2} + xe + f)^{2}} + \frac{2ax^{2}e^{2}}{(dx^{2} + xe + f)^{3}} - \frac{4axe}{(dx^{2} + xe + f)^{2}} + \frac{2a}{(dx^{2} + xe + f)} + \frac{8bdx^{2}e}{(dx^{2} + xe + f)^{3}} + \frac{8bd^{2}x^{3}}{(dx^{2} + xe + f)^{3}} - \frac{6bdx}{(dx^{2} + xe + f)^{2}} + \frac{2bxe^{2}}{(dx^{2} + xe + f)^{3}} - \frac{2be}{(dx^{2} + xe + f)^{2}} + \frac{8cdxe}{(dx^{2} + xe + f)^{3}} + \frac{8cd^{2}x^{2}}{(dx^{2} + xe + f)^{3}} - \frac{2cd}{(dx^{2} + xe + f)^{2}} + \frac{2ce^{2}}{(dx^{2} + xe + f)^{3}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{8adx^{3}e}{(dx^{2} + xe + f)^{3}} + \frac{8ad^{2}x^{4}}{(dx^{2} + xe + f)^{3}} - \frac{10adx^{2}}{(dx^{2} + xe + f)^{2}} + \frac{2ax^{2}e^{2}}{(dx^{2} + xe + f)^{3}} - \frac{4axe}{(dx^{2} + xe + f)^{2}} + \frac{2a}{(dx^{2} + xe + f)} + \frac{8bdx^{2}e}{(dx^{2} + xe + f)^{3}} + \frac{8bd^{2}x^{3}}{(dx^{2} + xe + f)^{3}} - \frac{6bdx}{(dx^{2} + xe + f)^{2}} + \frac{2bxe^{2}}{(dx^{2} + xe + f)^{3}} - \frac{2be}{(dx^{2} + xe + f)^{2}} + \frac{8cdxe}{(dx^{2} + xe + f)^{3}} + \frac{8cd^{2}x^{2}}{(dx^{2} + xe + f)^{3}} - \frac{2cd}{(dx^{2} + xe + f)^{2}} + \frac{2ce^{2}}{(dx^{2} + xe + f)^{3}}\right)}{dx}\\=&8(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})adx^{3}e + \frac{8ad*3x^{2}e}{(dx^{2} + xe + f)^{3}} + \frac{8adx^{3}*0}{(dx^{2} + xe + f)^{3}} + 8(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})ad^{2}x^{4} + \frac{8ad^{2}*4x^{3}}{(dx^{2} + xe + f)^{3}} - 10(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})adx^{2} - \frac{10ad*2x}{(dx^{2} + xe + f)^{2}} + 2(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})ax^{2}e^{2} + \frac{2a*2xe^{2}}{(dx^{2} + xe + f)^{3}} + \frac{2ax^{2}*2e*0}{(dx^{2} + xe + f)^{3}} - 4(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})axe - \frac{4ae}{(dx^{2} + xe + f)^{2}} - \frac{4ax*0}{(dx^{2} + xe + f)^{2}} + 2(\frac{-(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{2}})a + 0 + 8(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})bdx^{2}e + \frac{8bd*2xe}{(dx^{2} + xe + f)^{3}} + \frac{8bdx^{2}*0}{(dx^{2} + xe + f)^{3}} + 8(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})bd^{2}x^{3} + \frac{8bd^{2}*3x^{2}}{(dx^{2} + xe + f)^{3}} - 6(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})bdx - \frac{6bd}{(dx^{2} + xe + f)^{2}} + 2(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})bxe^{2} + \frac{2be^{2}}{(dx^{2} + xe + f)^{3}} + \frac{2bx*2e*0}{(dx^{2} + xe + f)^{3}} - 2(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})be - \frac{2b*0}{(dx^{2} + xe + f)^{2}} + 8(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})cdxe + \frac{8cde}{(dx^{2} + xe + f)^{3}} + \frac{8cdx*0}{(dx^{2} + xe + f)^{3}} + 8(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})cd^{2}x^{2} + \frac{8cd^{2}*2x}{(dx^{2} + xe + f)^{3}} - 2(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})cd + 0 + 2(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})ce^{2} + \frac{2c*2e*0}{(dx^{2} + xe + f)^{3}}\\=& - \frac{72ad^{2}x^{4}e}{(dx^{2} + xe + f)^{4}} - \frac{36adx^{3}e^{2}}{(dx^{2} + xe + f)^{4}} + \frac{60adx^{2}e}{(dx^{2} + xe + f)^{3}} - \frac{48ad^{3}x^{5}}{(dx^{2} + xe + f)^{4}} + \frac{72ad^{2}x^{3}}{(dx^{2} + xe + f)^{3}} - \frac{24adx}{(dx^{2} + xe + f)^{2}} - \frac{6ax^{2}e^{3}}{(dx^{2} + xe + f)^{4}} + \frac{12axe^{2}}{(dx^{2} + xe + f)^{3}} - \frac{6ae}{(dx^{2} + xe + f)^{2}} - \frac{72bd^{2}x^{3}e}{(dx^{2} + xe + f)^{4}} - \frac{36bdx^{2}e^{2}}{(dx^{2} + xe + f)^{4}} + \frac{36bdxe}{(dx^{2} + xe + f)^{3}} - \frac{48bd^{3}x^{4}}{(dx^{2} + xe + f)^{4}} + \frac{48bd^{2}x^{2}}{(dx^{2} + xe + f)^{3}} - \frac{6bd}{(dx^{2} + xe + f)^{2}} - \frac{6bxe^{3}}{(dx^{2} + xe + f)^{4}} + \frac{6be^{2}}{(dx^{2} + xe + f)^{3}} - \frac{72cd^{2}x^{2}e}{(dx^{2} + xe + f)^{4}} - \frac{36cdxe^{2}}{(dx^{2} + xe + f)^{4}} + \frac{12cde}{(dx^{2} + xe + f)^{3}} - \frac{48cd^{3}x^{3}}{(dx^{2} + xe + f)^{4}} + \frac{24cd^{2}x}{(dx^{2} + xe + f)^{3}} - \frac{6ce^{3}}{(dx^{2} + xe + f)^{4}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - \frac{72ad^{2}x^{4}e}{(dx^{2} + xe + f)^{4}} - \frac{36adx^{3}e^{2}}{(dx^{2} + xe + f)^{4}} + \frac{60adx^{2}e}{(dx^{2} + xe + f)^{3}} - \frac{48ad^{3}x^{5}}{(dx^{2} + xe + f)^{4}} + \frac{72ad^{2}x^{3}}{(dx^{2} + xe + f)^{3}} - \frac{24adx}{(dx^{2} + xe + f)^{2}} - \frac{6ax^{2}e^{3}}{(dx^{2} + xe + f)^{4}} + \frac{12axe^{2}}{(dx^{2} + xe + f)^{3}} - \frac{6ae}{(dx^{2} + xe + f)^{2}} - \frac{72bd^{2}x^{3}e}{(dx^{2} + xe + f)^{4}} - \frac{36bdx^{2}e^{2}}{(dx^{2} + xe + f)^{4}} + \frac{36bdxe}{(dx^{2} + xe + f)^{3}} - \frac{48bd^{3}x^{4}}{(dx^{2} + xe + f)^{4}} + \frac{48bd^{2}x^{2}}{(dx^{2} + xe + f)^{3}} - \frac{6bd}{(dx^{2} + xe + f)^{2}} - \frac{6bxe^{3}}{(dx^{2} + xe + f)^{4}} + \frac{6be^{2}}{(dx^{2} + xe + f)^{3}} - \frac{72cd^{2}x^{2}e}{(dx^{2} + xe + f)^{4}} - \frac{36cdxe^{2}}{(dx^{2} + xe + f)^{4}} + \frac{12cde}{(dx^{2} + xe + f)^{3}} - \frac{48cd^{3}x^{3}}{(dx^{2} + xe + f)^{4}} + \frac{24cd^{2}x}{(dx^{2} + xe + f)^{3}} - \frac{6ce^{3}}{(dx^{2} + xe + f)^{4}}\right)}{dx}\\=& - 72(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})ad^{2}x^{4}e - \frac{72ad^{2}*4x^{3}e}{(dx^{2} + xe + f)^{4}} - \frac{72ad^{2}x^{4}*0}{(dx^{2} + xe + f)^{4}} - 36(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})adx^{3}e^{2} - \frac{36ad*3x^{2}e^{2}}{(dx^{2} + xe + f)^{4}} - \frac{36adx^{3}*2e*0}{(dx^{2} + xe + f)^{4}} + 60(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})adx^{2}e + \frac{60ad*2xe}{(dx^{2} + xe + f)^{3}} + \frac{60adx^{2}*0}{(dx^{2} + xe + f)^{3}} - 48(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})ad^{3}x^{5} - \frac{48ad^{3}*5x^{4}}{(dx^{2} + xe + f)^{4}} + 72(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})ad^{2}x^{3} + \frac{72ad^{2}*3x^{2}}{(dx^{2} + xe + f)^{3}} - 24(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})adx - \frac{24ad}{(dx^{2} + xe + f)^{2}} - 6(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})ax^{2}e^{3} - \frac{6a*2xe^{3}}{(dx^{2} + xe + f)^{4}} - \frac{6ax^{2}*3e^{2}*0}{(dx^{2} + xe + f)^{4}} + 12(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})axe^{2} + \frac{12ae^{2}}{(dx^{2} + xe + f)^{3}} + \frac{12ax*2e*0}{(dx^{2} + xe + f)^{3}} - 6(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})ae - \frac{6a*0}{(dx^{2} + xe + f)^{2}} - 72(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})bd^{2}x^{3}e - \frac{72bd^{2}*3x^{2}e}{(dx^{2} + xe + f)^{4}} - \frac{72bd^{2}x^{3}*0}{(dx^{2} + xe + f)^{4}} - 36(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})bdx^{2}e^{2} - \frac{36bd*2xe^{2}}{(dx^{2} + xe + f)^{4}} - \frac{36bdx^{2}*2e*0}{(dx^{2} + xe + f)^{4}} + 36(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})bdxe + \frac{36bde}{(dx^{2} + xe + f)^{3}} + \frac{36bdx*0}{(dx^{2} + xe + f)^{3}} - 48(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})bd^{3}x^{4} - \frac{48bd^{3}*4x^{3}}{(dx^{2} + xe + f)^{4}} + 48(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})bd^{2}x^{2} + \frac{48bd^{2}*2x}{(dx^{2} + xe + f)^{3}} - 6(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})bd + 0 - 6(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})bxe^{3} - \frac{6be^{3}}{(dx^{2} + xe + f)^{4}} - \frac{6bx*3e^{2}*0}{(dx^{2} + xe + f)^{4}} + 6(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})be^{2} + \frac{6b*2e*0}{(dx^{2} + xe + f)^{3}} - 72(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})cd^{2}x^{2}e - \frac{72cd^{2}*2xe}{(dx^{2} + xe + f)^{4}} - \frac{72cd^{2}x^{2}*0}{(dx^{2} + xe + f)^{4}} - 36(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})cdxe^{2} - \frac{36cde^{2}}{(dx^{2} + xe + f)^{4}} - \frac{36cdx*2e*0}{(dx^{2} + xe + f)^{4}} + 12(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})cde + \frac{12cd*0}{(dx^{2} + xe + f)^{3}} - 48(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})cd^{3}x^{3} - \frac{48cd^{3}*3x^{2}}{(dx^{2} + xe + f)^{4}} + 24(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})cd^{2}x + \frac{24cd^{2}}{(dx^{2} + xe + f)^{3}} - 6(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})ce^{3} - \frac{6c*3e^{2}*0}{(dx^{2} + xe + f)^{4}}\\=&\frac{768ad^{3}x^{5}e}{(dx^{2} + xe + f)^{5}} + \frac{576ad^{2}x^{4}e^{2}}{(dx^{2} + xe + f)^{5}} - \frac{864ad^{2}x^{3}e}{(dx^{2} + xe + f)^{4}} + \frac{192adx^{3}e^{3}}{(dx^{2} + xe + f)^{5}} - \frac{360adx^{2}e^{2}}{(dx^{2} + xe + f)^{4}} + \frac{192adxe}{(dx^{2} + xe + f)^{3}} + \frac{384ad^{4}x^{6}}{(dx^{2} + xe + f)^{5}} - \frac{672ad^{3}x^{4}}{(dx^{2} + xe + f)^{4}} + \frac{312ad^{2}x^{2}}{(dx^{2} + xe + f)^{3}} - \frac{24ad}{(dx^{2} + xe + f)^{2}} + \frac{24ax^{2}e^{4}}{(dx^{2} + xe + f)^{5}} - \frac{48axe^{3}}{(dx^{2} + xe + f)^{4}} + \frac{24ae^{2}}{(dx^{2} + xe + f)^{3}} + \frac{768bd^{3}x^{4}e}{(dx^{2} + xe + f)^{5}} + \frac{576bd^{2}x^{3}e^{2}}{(dx^{2} + xe + f)^{5}} - \frac{576bd^{2}x^{2}e}{(dx^{2} + xe + f)^{4}} - \frac{216bdxe^{2}}{(dx^{2} + xe + f)^{4}} + \frac{192bdx^{2}e^{3}}{(dx^{2} + xe + f)^{5}} + \frac{48bde}{(dx^{2} + xe + f)^{3}} + \frac{384bd^{4}x^{5}}{(dx^{2} + xe + f)^{5}} - \frac{480bd^{3}x^{3}}{(dx^{2} + xe + f)^{4}} + \frac{120bd^{2}x}{(dx^{2} + xe + f)^{3}} + \frac{24bxe^{4}}{(dx^{2} + xe + f)^{5}} - \frac{24be^{3}}{(dx^{2} + xe + f)^{4}} + \frac{768cd^{3}x^{3}e}{(dx^{2} + xe + f)^{5}} + \frac{576cd^{2}x^{2}e^{2}}{(dx^{2} + xe + f)^{5}} - \frac{288cd^{2}xe}{(dx^{2} + xe + f)^{4}} + \frac{192cdxe^{3}}{(dx^{2} + xe + f)^{5}} - \frac{72cde^{2}}{(dx^{2} + xe + f)^{4}} - \frac{288cd^{3}x^{2}}{(dx^{2} + xe + f)^{4}} + \frac{384cd^{4}x^{4}}{(dx^{2} + xe + f)^{5}} + \frac{24cd^{2}}{(dx^{2} + xe + f)^{3}} + \frac{24ce^{4}}{(dx^{2} + xe + f)^{5}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{(axx + bx + c)}{(dxx + xe + f)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{ax^{2}}{(dx^{2} + xe + f)} + \frac{bx}{(dx^{2} + xe + f)} + \frac{c}{(dx^{2} + xe + f)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{ax^{2}}{(dx^{2} + xe + f)} + \frac{bx}{(dx^{2} + xe + f)} + \frac{c}{(dx^{2} + xe + f)}\right)}{dx}\\=&(\frac{-(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{2}})ax^{2} + \frac{a*2x}{(dx^{2} + xe + f)} + (\frac{-(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{2}})bx + \frac{b}{(dx^{2} + xe + f)} + (\frac{-(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{2}})c + 0\\=&\frac{-2adx^{3}}{(dx^{2} + xe + f)^{2}} - \frac{ax^{2}e}{(dx^{2} + xe + f)^{2}} + \frac{2ax}{(dx^{2} + xe + f)} - \frac{2bdx^{2}}{(dx^{2} + xe + f)^{2}} - \frac{bxe}{(dx^{2} + xe + f)^{2}} + \frac{b}{(dx^{2} + xe + f)} - \frac{2cdx}{(dx^{2} + xe + f)^{2}} - \frac{ce}{(dx^{2} + xe + f)^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2adx^{3}}{(dx^{2} + xe + f)^{2}} - \frac{ax^{2}e}{(dx^{2} + xe + f)^{2}} + \frac{2ax}{(dx^{2} + xe + f)} - \frac{2bdx^{2}}{(dx^{2} + xe + f)^{2}} - \frac{bxe}{(dx^{2} + xe + f)^{2}} + \frac{b}{(dx^{2} + xe + f)} - \frac{2cdx}{(dx^{2} + xe + f)^{2}} - \frac{ce}{(dx^{2} + xe + f)^{2}}\right)}{dx}\\=&-2(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})adx^{3} - \frac{2ad*3x^{2}}{(dx^{2} + xe + f)^{2}} - (\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})ax^{2}e - \frac{a*2xe}{(dx^{2} + xe + f)^{2}} - \frac{ax^{2}*0}{(dx^{2} + xe + f)^{2}} + 2(\frac{-(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{2}})ax + \frac{2a}{(dx^{2} + xe + f)} - 2(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})bdx^{2} - \frac{2bd*2x}{(dx^{2} + xe + f)^{2}} - (\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})bxe - \frac{be}{(dx^{2} + xe + f)^{2}} - \frac{bx*0}{(dx^{2} + xe + f)^{2}} + (\frac{-(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{2}})b + 0 - 2(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})cdx - \frac{2cd}{(dx^{2} + xe + f)^{2}} - (\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})ce - \frac{c*0}{(dx^{2} + xe + f)^{2}}\\=&\frac{8adx^{3}e}{(dx^{2} + xe + f)^{3}} + \frac{8ad^{2}x^{4}}{(dx^{2} + xe + f)^{3}} - \frac{10adx^{2}}{(dx^{2} + xe + f)^{2}} + \frac{2ax^{2}e^{2}}{(dx^{2} + xe + f)^{3}} - \frac{4axe}{(dx^{2} + xe + f)^{2}} + \frac{2a}{(dx^{2} + xe + f)} + \frac{8bdx^{2}e}{(dx^{2} + xe + f)^{3}} + \frac{8bd^{2}x^{3}}{(dx^{2} + xe + f)^{3}} - \frac{6bdx}{(dx^{2} + xe + f)^{2}} + \frac{2bxe^{2}}{(dx^{2} + xe + f)^{3}} - \frac{2be}{(dx^{2} + xe + f)^{2}} + \frac{8cdxe}{(dx^{2} + xe + f)^{3}} + \frac{8cd^{2}x^{2}}{(dx^{2} + xe + f)^{3}} - \frac{2cd}{(dx^{2} + xe + f)^{2}} + \frac{2ce^{2}}{(dx^{2} + xe + f)^{3}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{8adx^{3}e}{(dx^{2} + xe + f)^{3}} + \frac{8ad^{2}x^{4}}{(dx^{2} + xe + f)^{3}} - \frac{10adx^{2}}{(dx^{2} + xe + f)^{2}} + \frac{2ax^{2}e^{2}}{(dx^{2} + xe + f)^{3}} - \frac{4axe}{(dx^{2} + xe + f)^{2}} + \frac{2a}{(dx^{2} + xe + f)} + \frac{8bdx^{2}e}{(dx^{2} + xe + f)^{3}} + \frac{8bd^{2}x^{3}}{(dx^{2} + xe + f)^{3}} - \frac{6bdx}{(dx^{2} + xe + f)^{2}} + \frac{2bxe^{2}}{(dx^{2} + xe + f)^{3}} - \frac{2be}{(dx^{2} + xe + f)^{2}} + \frac{8cdxe}{(dx^{2} + xe + f)^{3}} + \frac{8cd^{2}x^{2}}{(dx^{2} + xe + f)^{3}} - \frac{2cd}{(dx^{2} + xe + f)^{2}} + \frac{2ce^{2}}{(dx^{2} + xe + f)^{3}}\right)}{dx}\\=&8(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})adx^{3}e + \frac{8ad*3x^{2}e}{(dx^{2} + xe + f)^{3}} + \frac{8adx^{3}*0}{(dx^{2} + xe + f)^{3}} + 8(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})ad^{2}x^{4} + \frac{8ad^{2}*4x^{3}}{(dx^{2} + xe + f)^{3}} - 10(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})adx^{2} - \frac{10ad*2x}{(dx^{2} + xe + f)^{2}} + 2(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})ax^{2}e^{2} + \frac{2a*2xe^{2}}{(dx^{2} + xe + f)^{3}} + \frac{2ax^{2}*2e*0}{(dx^{2} + xe + f)^{3}} - 4(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})axe - \frac{4ae}{(dx^{2} + xe + f)^{2}} - \frac{4ax*0}{(dx^{2} + xe + f)^{2}} + 2(\frac{-(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{2}})a + 0 + 8(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})bdx^{2}e + \frac{8bd*2xe}{(dx^{2} + xe + f)^{3}} + \frac{8bdx^{2}*0}{(dx^{2} + xe + f)^{3}} + 8(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})bd^{2}x^{3} + \frac{8bd^{2}*3x^{2}}{(dx^{2} + xe + f)^{3}} - 6(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})bdx - \frac{6bd}{(dx^{2} + xe + f)^{2}} + 2(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})bxe^{2} + \frac{2be^{2}}{(dx^{2} + xe + f)^{3}} + \frac{2bx*2e*0}{(dx^{2} + xe + f)^{3}} - 2(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})be - \frac{2b*0}{(dx^{2} + xe + f)^{2}} + 8(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})cdxe + \frac{8cde}{(dx^{2} + xe + f)^{3}} + \frac{8cdx*0}{(dx^{2} + xe + f)^{3}} + 8(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})cd^{2}x^{2} + \frac{8cd^{2}*2x}{(dx^{2} + xe + f)^{3}} - 2(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})cd + 0 + 2(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})ce^{2} + \frac{2c*2e*0}{(dx^{2} + xe + f)^{3}}\\=& - \frac{72ad^{2}x^{4}e}{(dx^{2} + xe + f)^{4}} - \frac{36adx^{3}e^{2}}{(dx^{2} + xe + f)^{4}} + \frac{60adx^{2}e}{(dx^{2} + xe + f)^{3}} - \frac{48ad^{3}x^{5}}{(dx^{2} + xe + f)^{4}} + \frac{72ad^{2}x^{3}}{(dx^{2} + xe + f)^{3}} - \frac{24adx}{(dx^{2} + xe + f)^{2}} - \frac{6ax^{2}e^{3}}{(dx^{2} + xe + f)^{4}} + \frac{12axe^{2}}{(dx^{2} + xe + f)^{3}} - \frac{6ae}{(dx^{2} + xe + f)^{2}} - \frac{72bd^{2}x^{3}e}{(dx^{2} + xe + f)^{4}} - \frac{36bdx^{2}e^{2}}{(dx^{2} + xe + f)^{4}} + \frac{36bdxe}{(dx^{2} + xe + f)^{3}} - \frac{48bd^{3}x^{4}}{(dx^{2} + xe + f)^{4}} + \frac{48bd^{2}x^{2}}{(dx^{2} + xe + f)^{3}} - \frac{6bd}{(dx^{2} + xe + f)^{2}} - \frac{6bxe^{3}}{(dx^{2} + xe + f)^{4}} + \frac{6be^{2}}{(dx^{2} + xe + f)^{3}} - \frac{72cd^{2}x^{2}e}{(dx^{2} + xe + f)^{4}} - \frac{36cdxe^{2}}{(dx^{2} + xe + f)^{4}} + \frac{12cde}{(dx^{2} + xe + f)^{3}} - \frac{48cd^{3}x^{3}}{(dx^{2} + xe + f)^{4}} + \frac{24cd^{2}x}{(dx^{2} + xe + f)^{3}} - \frac{6ce^{3}}{(dx^{2} + xe + f)^{4}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - \frac{72ad^{2}x^{4}e}{(dx^{2} + xe + f)^{4}} - \frac{36adx^{3}e^{2}}{(dx^{2} + xe + f)^{4}} + \frac{60adx^{2}e}{(dx^{2} + xe + f)^{3}} - \frac{48ad^{3}x^{5}}{(dx^{2} + xe + f)^{4}} + \frac{72ad^{2}x^{3}}{(dx^{2} + xe + f)^{3}} - \frac{24adx}{(dx^{2} + xe + f)^{2}} - \frac{6ax^{2}e^{3}}{(dx^{2} + xe + f)^{4}} + \frac{12axe^{2}}{(dx^{2} + xe + f)^{3}} - \frac{6ae}{(dx^{2} + xe + f)^{2}} - \frac{72bd^{2}x^{3}e}{(dx^{2} + xe + f)^{4}} - \frac{36bdx^{2}e^{2}}{(dx^{2} + xe + f)^{4}} + \frac{36bdxe}{(dx^{2} + xe + f)^{3}} - \frac{48bd^{3}x^{4}}{(dx^{2} + xe + f)^{4}} + \frac{48bd^{2}x^{2}}{(dx^{2} + xe + f)^{3}} - \frac{6bd}{(dx^{2} + xe + f)^{2}} - \frac{6bxe^{3}}{(dx^{2} + xe + f)^{4}} + \frac{6be^{2}}{(dx^{2} + xe + f)^{3}} - \frac{72cd^{2}x^{2}e}{(dx^{2} + xe + f)^{4}} - \frac{36cdxe^{2}}{(dx^{2} + xe + f)^{4}} + \frac{12cde}{(dx^{2} + xe + f)^{3}} - \frac{48cd^{3}x^{3}}{(dx^{2} + xe + f)^{4}} + \frac{24cd^{2}x}{(dx^{2} + xe + f)^{3}} - \frac{6ce^{3}}{(dx^{2} + xe + f)^{4}}\right)}{dx}\\=& - 72(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})ad^{2}x^{4}e - \frac{72ad^{2}*4x^{3}e}{(dx^{2} + xe + f)^{4}} - \frac{72ad^{2}x^{4}*0}{(dx^{2} + xe + f)^{4}} - 36(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})adx^{3}e^{2} - \frac{36ad*3x^{2}e^{2}}{(dx^{2} + xe + f)^{4}} - \frac{36adx^{3}*2e*0}{(dx^{2} + xe + f)^{4}} + 60(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})adx^{2}e + \frac{60ad*2xe}{(dx^{2} + xe + f)^{3}} + \frac{60adx^{2}*0}{(dx^{2} + xe + f)^{3}} - 48(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})ad^{3}x^{5} - \frac{48ad^{3}*5x^{4}}{(dx^{2} + xe + f)^{4}} + 72(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})ad^{2}x^{3} + \frac{72ad^{2}*3x^{2}}{(dx^{2} + xe + f)^{3}} - 24(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})adx - \frac{24ad}{(dx^{2} + xe + f)^{2}} - 6(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})ax^{2}e^{3} - \frac{6a*2xe^{3}}{(dx^{2} + xe + f)^{4}} - \frac{6ax^{2}*3e^{2}*0}{(dx^{2} + xe + f)^{4}} + 12(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})axe^{2} + \frac{12ae^{2}}{(dx^{2} + xe + f)^{3}} + \frac{12ax*2e*0}{(dx^{2} + xe + f)^{3}} - 6(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})ae - \frac{6a*0}{(dx^{2} + xe + f)^{2}} - 72(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})bd^{2}x^{3}e - \frac{72bd^{2}*3x^{2}e}{(dx^{2} + xe + f)^{4}} - \frac{72bd^{2}x^{3}*0}{(dx^{2} + xe + f)^{4}} - 36(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})bdx^{2}e^{2} - \frac{36bd*2xe^{2}}{(dx^{2} + xe + f)^{4}} - \frac{36bdx^{2}*2e*0}{(dx^{2} + xe + f)^{4}} + 36(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})bdxe + \frac{36bde}{(dx^{2} + xe + f)^{3}} + \frac{36bdx*0}{(dx^{2} + xe + f)^{3}} - 48(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})bd^{3}x^{4} - \frac{48bd^{3}*4x^{3}}{(dx^{2} + xe + f)^{4}} + 48(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})bd^{2}x^{2} + \frac{48bd^{2}*2x}{(dx^{2} + xe + f)^{3}} - 6(\frac{-2(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{3}})bd + 0 - 6(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})bxe^{3} - \frac{6be^{3}}{(dx^{2} + xe + f)^{4}} - \frac{6bx*3e^{2}*0}{(dx^{2} + xe + f)^{4}} + 6(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})be^{2} + \frac{6b*2e*0}{(dx^{2} + xe + f)^{3}} - 72(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})cd^{2}x^{2}e - \frac{72cd^{2}*2xe}{(dx^{2} + xe + f)^{4}} - \frac{72cd^{2}x^{2}*0}{(dx^{2} + xe + f)^{4}} - 36(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})cdxe^{2} - \frac{36cde^{2}}{(dx^{2} + xe + f)^{4}} - \frac{36cdx*2e*0}{(dx^{2} + xe + f)^{4}} + 12(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})cde + \frac{12cd*0}{(dx^{2} + xe + f)^{3}} - 48(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})cd^{3}x^{3} - \frac{48cd^{3}*3x^{2}}{(dx^{2} + xe + f)^{4}} + 24(\frac{-3(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{4}})cd^{2}x + \frac{24cd^{2}}{(dx^{2} + xe + f)^{3}} - 6(\frac{-4(d*2x + e + x*0 + 0)}{(dx^{2} + xe + f)^{5}})ce^{3} - \frac{6c*3e^{2}*0}{(dx^{2} + xe + f)^{4}}\\=&\frac{768ad^{3}x^{5}e}{(dx^{2} + xe + f)^{5}} + \frac{576ad^{2}x^{4}e^{2}}{(dx^{2} + xe + f)^{5}} - \frac{864ad^{2}x^{3}e}{(dx^{2} + xe + f)^{4}} + \frac{192adx^{3}e^{3}}{(dx^{2} + xe + f)^{5}} - \frac{360adx^{2}e^{2}}{(dx^{2} + xe + f)^{4}} + \frac{192adxe}{(dx^{2} + xe + f)^{3}} + \frac{384ad^{4}x^{6}}{(dx^{2} + xe + f)^{5}} - \frac{672ad^{3}x^{4}}{(dx^{2} + xe + f)^{4}} + \frac{312ad^{2}x^{2}}{(dx^{2} + xe + f)^{3}} - \frac{24ad}{(dx^{2} + xe + f)^{2}} + \frac{24ax^{2}e^{4}}{(dx^{2} + xe + f)^{5}} - \frac{48axe^{3}}{(dx^{2} + xe + f)^{4}} + \frac{24ae^{2}}{(dx^{2} + xe + f)^{3}} + \frac{768bd^{3}x^{4}e}{(dx^{2} + xe + f)^{5}} + \frac{576bd^{2}x^{3}e^{2}}{(dx^{2} + xe + f)^{5}} - \frac{576bd^{2}x^{2}e}{(dx^{2} + xe + f)^{4}} - \frac{216bdxe^{2}}{(dx^{2} + xe + f)^{4}} + \frac{192bdx^{2}e^{3}}{(dx^{2} + xe + f)^{5}} + \frac{48bde}{(dx^{2} + xe + f)^{3}} + \frac{384bd^{4}x^{5}}{(dx^{2} + xe + f)^{5}} - \frac{480bd^{3}x^{3}}{(dx^{2} + xe + f)^{4}} + \frac{120bd^{2}x}{(dx^{2} + xe + f)^{3}} + \frac{24bxe^{4}}{(dx^{2} + xe + f)^{5}} - \frac{24be^{3}}{(dx^{2} + xe + f)^{4}} + \frac{768cd^{3}x^{3}e}{(dx^{2} + xe + f)^{5}} + \frac{576cd^{2}x^{2}e^{2}}{(dx^{2} + xe + f)^{5}} - \frac{288cd^{2}xe}{(dx^{2} + xe + f)^{4}} + \frac{192cdxe^{3}}{(dx^{2} + xe + f)^{5}} - \frac{72cde^{2}}{(dx^{2} + xe + f)^{4}} - \frac{288cd^{3}x^{2}}{(dx^{2} + xe + f)^{4}} + \frac{384cd^{4}x^{4}}{(dx^{2} + xe + f)^{5}} + \frac{24cd^{2}}{(dx^{2} + xe + f)^{3}} + \frac{24ce^{4}}{(dx^{2} + xe + f)^{5}}\\ \end{split}\end{equation} \]