Derivative function：
Enter an original function (that is, the function to be derived), then set the variable to be derived and the order of the derivative, and click the "Next" button to obtain the derivative function of the corresponding order of the function.
Note that the input function supports mathematical functions and other constants.
Enter an original function (that is, the function to be derived), then set the variable to be derived and the order of the derivative, and click the "Next" button to obtain the derivative function of the corresponding order of the function.
Note that the input function supports mathematical functions and other constants.
Current location：Derivative function > Derivative function calculation history > Answer
There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{{x}^{6}}{(x + {x}^{2} + {x}^{3} + {x}^{4} + {x}^{5})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})}\right)}{dx}\\=&(\frac{-(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}})x^{6} + \frac{6x^{5}}{(x + x^{2} + x^{3} + x^{4} + x^{5})}\\=&\frac{-2x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - \frac{3x^{8}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - \frac{4x^{9}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - \frac{5x^{10}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - \frac{x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + \frac{6x^{5}}{(x + x^{2} + x^{3} + x^{4} + x^{5})}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - \frac{3x^{8}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - \frac{4x^{9}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - \frac{5x^{10}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - \frac{x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + \frac{6x^{5}}{(x + x^{2} + x^{3} + x^{4} + x^{5})}\right)}{dx}\\=&-2(\frac{-2(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}})x^{7} - \frac{2*7x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - 3(\frac{-2(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}})x^{8} - \frac{3*8x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - 4(\frac{-2(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}})x^{9} - \frac{4*9x^{8}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - 5(\frac{-2(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}})x^{10} - \frac{5*10x^{9}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - (\frac{-2(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}})x^{6} - \frac{6x^{5}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + 6(\frac{-(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}})x^{5} + \frac{6*5x^{4}}{(x + x^{2} + x^{3} + x^{4} + x^{5})}\\=&\frac{20x^{8}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} + \frac{40x^{9}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} + \frac{70x^{10}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} + \frac{88x^{11}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{26x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + \frac{92x^{12}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{42x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + \frac{80x^{13}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{60x^{8}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + \frac{50x^{14}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} + \frac{8x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{80x^{9}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + \frac{2x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{12x^{5}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + \frac{30x^{4}}{(x + x^{2} + x^{3} + x^{4} + x^{5})}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{20x^{8}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} + \frac{40x^{9}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} + \frac{70x^{10}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} + \frac{88x^{11}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{26x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + \frac{92x^{12}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{42x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + \frac{80x^{13}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{60x^{8}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + \frac{50x^{14}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} + \frac{8x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{80x^{9}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + \frac{2x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{12x^{5}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + \frac{30x^{4}}{(x + x^{2} + x^{3} + x^{4} + x^{5})}\right)}{dx}\\=&20(\frac{-3(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}})x^{8} + \frac{20*8x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} + 40(\frac{-3(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}})x^{9} + \frac{40*9x^{8}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} + 70(\frac{-3(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}})x^{10} + \frac{70*10x^{9}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} + 88(\frac{-3(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}})x^{11} + \frac{88*11x^{10}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - 26(\frac{-2(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}})x^{6} - \frac{26*6x^{5}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + 92(\frac{-3(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}})x^{12} + \frac{92*12x^{11}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - 42(\frac{-2(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}})x^{7} - \frac{42*7x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + 80(\frac{-3(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}})x^{13} + \frac{80*13x^{12}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - 60(\frac{-2(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}})x^{8} - \frac{60*8x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + 50(\frac{-3(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}})x^{14} + \frac{50*14x^{13}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} + 8(\frac{-3(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}})x^{7} + \frac{8*7x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - 80(\frac{-2(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}})x^{9} - \frac{80*9x^{8}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + 2(\frac{-3(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}})x^{6} + \frac{2*6x^{5}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - 12(\frac{-2(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}})x^{5} - \frac{12*5x^{4}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + 30(\frac{-(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}})x^{4} + \frac{30*4x^{3}}{(x + x^{2} + x^{3} + x^{4} + x^{5})}\\=&\frac{-336x^{9}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} - \frac{756x^{10}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} - \frac{1404x^{11}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} - \frac{2214x^{12}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{420x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{3024x^{13}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{900x^{8}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{3564x^{14}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{1680x^{9}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{3444x^{15}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} - \frac{2790x^{16}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{2244x^{10}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{216x^{5}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - \frac{1800x^{17}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{2484x^{11}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{384x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - \frac{750x^{18}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} - \frac{126x^{8}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{2280x^{12}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{600x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - \frac{36x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{1500x^{13}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} + \frac{156x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{870x^{8}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - \frac{6x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{36x^{5}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{90x^{4}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + \frac{120x^{3}}{(x + x^{2} + x^{3} + x^{4} + x^{5})}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-336x^{9}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} - \frac{756x^{10}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} - \frac{1404x^{11}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} - \frac{2214x^{12}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{420x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{3024x^{13}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{900x^{8}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{3564x^{14}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{1680x^{9}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{3444x^{15}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} - \frac{2790x^{16}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{2244x^{10}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{216x^{5}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - \frac{1800x^{17}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{2484x^{11}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{384x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - \frac{750x^{18}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} - \frac{126x^{8}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{2280x^{12}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{600x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - \frac{36x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{1500x^{13}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} + \frac{156x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{870x^{8}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - \frac{6x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{36x^{5}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{90x^{4}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + \frac{120x^{3}}{(x + x^{2} + x^{3} + x^{4} + x^{5})}\right)}{dx}\\=&-336(\frac{-4(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}})x^{9} - \frac{336*9x^{8}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} - 756(\frac{-4(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}})x^{10} - \frac{756*10x^{9}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} - 1404(\frac{-4(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}})x^{11} - \frac{1404*11x^{10}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} - 2214(\frac{-4(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}})x^{12} - \frac{2214*12x^{11}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + 420(\frac{-3(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}})x^{7} + \frac{420*7x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - 3024(\frac{-4(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}})x^{13} - \frac{3024*13x^{12}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + 900(\frac{-3(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}})x^{8} + \frac{900*8x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - 3564(\frac{-4(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}})x^{14} - \frac{3564*14x^{13}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + 1680(\frac{-3(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}})x^{9} + \frac{1680*9x^{8}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - 3444(\frac{-4(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}})x^{15} - \frac{3444*15x^{14}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} - 2790(\frac{-4(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}})x^{16} - \frac{2790*16x^{15}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + 2244(\frac{-3(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}})x^{10} + \frac{2244*10x^{9}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - 216(\frac{-2(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}})x^{5} - \frac{216*5x^{4}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - 1800(\frac{-4(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}})x^{17} - \frac{1800*17x^{16}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + 2484(\frac{-3(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}})x^{11} + \frac{2484*11x^{10}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - 384(\frac{-2(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}})x^{6} - \frac{384*6x^{5}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - 750(\frac{-4(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}})x^{18} - \frac{750*18x^{17}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} - 126(\frac{-4(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}})x^{8} - \frac{126*8x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + 2280(\frac{-3(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}})x^{12} + \frac{2280*12x^{11}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - 600(\frac{-2(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}})x^{7} - \frac{600*7x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - 36(\frac{-4(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}})x^{7} - \frac{36*7x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + 1500(\frac{-3(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}})x^{13} + \frac{1500*13x^{12}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} + 156(\frac{-3(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}})x^{6} + \frac{156*6x^{5}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - 870(\frac{-2(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}})x^{8} - \frac{870*8x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - 6(\frac{-4(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}})x^{6} - \frac{6*6x^{5}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + 36(\frac{-3(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}})x^{5} + \frac{36*5x^{4}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - 90(\frac{-2(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}})x^{4} - \frac{90*4x^{3}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + 120(\frac{-(1 + 2x + 3x^{2} + 4x^{3} + 5x^{4})}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}})x^{3} + \frac{120*3x^{2}}{(x + x^{2} + x^{3} + x^{4} + x^{5})}\\=&\frac{7920x^{10}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}} + \frac{18432x^{11}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}} + \frac{37056x^{12}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}} + \frac{65472x^{13}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}} - \frac{10080x^{8}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{102600x^{14}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}} - \frac{24192x^{9}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{142080x^{15}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}} - \frac{47736x^{10}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{174144x^{16}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}} + \frac{188352x^{17}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}} - \frac{79704x^{11}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{5112x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} + \frac{177264x^{18}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}} - \frac{114912x^{12}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{11952x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} + \frac{141120x^{19}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}} - \frac{142560x^{13}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{24192x^{8}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} + \frac{93600x^{20}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}} - \frac{144648x^{14}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{48000x^{21}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}} + \frac{15000x^{22}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}} - \frac{122760x^{15}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{34752x^{9}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{1320x^{4}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + \frac{2880x^{9}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}} + \frac{864x^{8}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}} - \frac{82800x^{16}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{41184x^{10}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{2664x^{5}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + \frac{192x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}} - \frac{36000x^{17}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} - \frac{3528x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{40320x^{11}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{4680x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} - \frac{936x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{28200x^{12}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} + \frac{1728x^{5}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{7560x^{7}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + \frac{24x^{6}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{5}} - \frac{144x^{5}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{4}} + \frac{360x^{4}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}} - \frac{480x^{3}}{(x + x^{2} + x^{3} + x^{4} + x^{5})^{2}} + \frac{360x^{2}}{(x + x^{2} + x^{3} + x^{4} + x^{5})}\\ \end{split}\end{equation} \]