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 3 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ \frac{({x}^{8} + 2{x}^{4} + 1)}{((1 + {a}^{2})x - a({x}^{2} + 1))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{x^{8}}{(x + a^{2}x - ax^{2} - a)} + \frac{2x^{4}}{(x + a^{2}x - ax^{2} - a)} + \frac{1}{(x + a^{2}x - ax^{2} - a)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x^{8}}{(x + a^{2}x - ax^{2} - a)} + \frac{2x^{4}}{(x + a^{2}x - ax^{2} - a)} + \frac{1}{(x + a^{2}x - ax^{2} - a)}\right)}{dx}\\=&(\frac{-(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{2}})x^{8} + \frac{8x^{7}}{(x + a^{2}x - ax^{2} - a)} + 2(\frac{-(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{2}})x^{4} + \frac{2*4x^{3}}{(x + a^{2}x - ax^{2} - a)} + (\frac{-(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{2}})\\=&\frac{2ax^{9}}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{4ax^{5}}{(x + a^{2}x - ax^{2} - a)^{2}} - \frac{x^{8}}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{8x^{7}}{(x + a^{2}x - ax^{2} - a)} - \frac{2a^{2}x^{4}}{(x + a^{2}x - ax^{2} - a)^{2}} - \frac{a^{2}x^{8}}{(x + a^{2}x - ax^{2} - a)^{2}} - \frac{2x^{4}}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{8x^{3}}{(x + a^{2}x - ax^{2} - a)} + \frac{2ax}{(x + a^{2}x - ax^{2} - a)^{2}} - \frac{a^{2}}{(x + a^{2}x - ax^{2} - a)^{2}} - \frac{1}{(x + a^{2}x - ax^{2} - a)^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{2ax^{9}}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{4ax^{5}}{(x + a^{2}x - ax^{2} - a)^{2}} - \frac{x^{8}}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{8x^{7}}{(x + a^{2}x - ax^{2} - a)} - \frac{2a^{2}x^{4}}{(x + a^{2}x - ax^{2} - a)^{2}} - \frac{a^{2}x^{8}}{(x + a^{2}x - ax^{2} - a)^{2}} - \frac{2x^{4}}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{8x^{3}}{(x + a^{2}x - ax^{2} - a)} + \frac{2ax}{(x + a^{2}x - ax^{2} - a)^{2}} - \frac{a^{2}}{(x + a^{2}x - ax^{2} - a)^{2}} - \frac{1}{(x + a^{2}x - ax^{2} - a)^{2}}\right)}{dx}\\=&2(\frac{-2(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{3}})ax^{9} + \frac{2a*9x^{8}}{(x + a^{2}x - ax^{2} - a)^{2}} + 4(\frac{-2(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{3}})ax^{5} + \frac{4a*5x^{4}}{(x + a^{2}x - ax^{2} - a)^{2}} - (\frac{-2(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{3}})x^{8} - \frac{8x^{7}}{(x + a^{2}x - ax^{2} - a)^{2}} + 8(\frac{-(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{2}})x^{7} + \frac{8*7x^{6}}{(x + a^{2}x - ax^{2} - a)} - 2(\frac{-2(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{3}})a^{2}x^{4} - \frac{2a^{2}*4x^{3}}{(x + a^{2}x - ax^{2} - a)^{2}} - (\frac{-2(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{3}})a^{2}x^{8} - \frac{a^{2}*8x^{7}}{(x + a^{2}x - ax^{2} - a)^{2}} - 2(\frac{-2(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{3}})x^{4} - \frac{2*4x^{3}}{(x + a^{2}x - ax^{2} - a)^{2}} + 8(\frac{-(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{2}})x^{3} + \frac{8*3x^{2}}{(x + a^{2}x - ax^{2} - a)} + 2(\frac{-2(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{3}})ax + \frac{2a}{(x + a^{2}x - ax^{2} - a)^{2}} - (\frac{-2(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{3}})a^{2} + 0 - (\frac{-2(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{3}})\\=&\frac{8a^{2}x^{10}}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{16a^{2}x^{6}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{8ax^{9}}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{34ax^{8}}{(x + a^{2}x - ax^{2} - a)^{2}} - \frac{16a^{3}x^{5}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{8a^{3}x^{9}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{16ax^{5}}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{36ax^{4}}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{4a^{2}x^{8}}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{2x^{8}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{16x^{7}}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{56x^{6}}{(x + a^{2}x - ax^{2} - a)} + \frac{8a^{2}x^{2}}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{8a^{2}x^{4}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{16a^{2}x^{3}}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{2a^{4}x^{8}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{16a^{2}x^{7}}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{4x^{4}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{16x^{3}}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{24x^{2}}{(x + a^{2}x - ax^{2} - a)} - \frac{8a^{3}x}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{4a^{4}x^{4}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{8ax}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{2a}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{2a^{4}}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{4a^{2}}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{2}{(x + a^{2}x - ax^{2} - a)^{3}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{8a^{2}x^{10}}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{16a^{2}x^{6}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{8ax^{9}}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{34ax^{8}}{(x + a^{2}x - ax^{2} - a)^{2}} - \frac{16a^{3}x^{5}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{8a^{3}x^{9}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{16ax^{5}}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{36ax^{4}}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{4a^{2}x^{8}}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{2x^{8}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{16x^{7}}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{56x^{6}}{(x + a^{2}x - ax^{2} - a)} + \frac{8a^{2}x^{2}}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{8a^{2}x^{4}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{16a^{2}x^{3}}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{2a^{4}x^{8}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{16a^{2}x^{7}}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{4x^{4}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{16x^{3}}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{24x^{2}}{(x + a^{2}x - ax^{2} - a)} - \frac{8a^{3}x}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{4a^{4}x^{4}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{8ax}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{2a}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{2a^{4}}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{4a^{2}}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{2}{(x + a^{2}x - ax^{2} - a)^{3}}\right)}{dx}\\=&8(\frac{-3(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{4}})a^{2}x^{10} + \frac{8a^{2}*10x^{9}}{(x + a^{2}x - ax^{2} - a)^{3}} + 16(\frac{-3(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{4}})a^{2}x^{6} + \frac{16a^{2}*6x^{5}}{(x + a^{2}x - ax^{2} - a)^{3}} - 8(\frac{-3(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{4}})ax^{9} - \frac{8a*9x^{8}}{(x + a^{2}x - ax^{2} - a)^{3}} + 34(\frac{-2(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{3}})ax^{8} + \frac{34a*8x^{7}}{(x + a^{2}x - ax^{2} - a)^{2}} - 16(\frac{-3(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{4}})a^{3}x^{5} - \frac{16a^{3}*5x^{4}}{(x + a^{2}x - ax^{2} - a)^{3}} - 8(\frac{-3(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{4}})a^{3}x^{9} - \frac{8a^{3}*9x^{8}}{(x + a^{2}x - ax^{2} - a)^{3}} - 16(\frac{-3(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{4}})ax^{5} - \frac{16a*5x^{4}}{(x + a^{2}x - ax^{2} - a)^{3}} + 36(\frac{-2(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{3}})ax^{4} + \frac{36a*4x^{3}}{(x + a^{2}x - ax^{2} - a)^{2}} + 4(\frac{-3(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{4}})a^{2}x^{8} + \frac{4a^{2}*8x^{7}}{(x + a^{2}x - ax^{2} - a)^{3}} + 2(\frac{-3(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{4}})x^{8} + \frac{2*8x^{7}}{(x + a^{2}x - ax^{2} - a)^{3}} - 16(\frac{-2(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{3}})x^{7} - \frac{16*7x^{6}}{(x + a^{2}x - ax^{2} - a)^{2}} + 56(\frac{-(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{2}})x^{6} + \frac{56*6x^{5}}{(x + a^{2}x - ax^{2} - a)} + 8(\frac{-3(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{4}})a^{2}x^{2} + \frac{8a^{2}*2x}{(x + a^{2}x - ax^{2} - a)^{3}} + 8(\frac{-3(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{4}})a^{2}x^{4} + \frac{8a^{2}*4x^{3}}{(x + a^{2}x - ax^{2} - a)^{3}} - 16(\frac{-2(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{3}})a^{2}x^{3} - \frac{16a^{2}*3x^{2}}{(x + a^{2}x - ax^{2} - a)^{2}} + 2(\frac{-3(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{4}})a^{4}x^{8} + \frac{2a^{4}*8x^{7}}{(x + a^{2}x - ax^{2} - a)^{3}} - 16(\frac{-2(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{3}})a^{2}x^{7} - \frac{16a^{2}*7x^{6}}{(x + a^{2}x - ax^{2} - a)^{2}} + 4(\frac{-3(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{4}})x^{4} + \frac{4*4x^{3}}{(x + a^{2}x - ax^{2} - a)^{3}} - 16(\frac{-2(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{3}})x^{3} - \frac{16*3x^{2}}{(x + a^{2}x - ax^{2} - a)^{2}} + 24(\frac{-(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{2}})x^{2} + \frac{24*2x}{(x + a^{2}x - ax^{2} - a)} - 8(\frac{-3(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{4}})a^{3}x - \frac{8a^{3}}{(x + a^{2}x - ax^{2} - a)^{3}} + 4(\frac{-3(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{4}})a^{4}x^{4} + \frac{4a^{4}*4x^{3}}{(x + a^{2}x - ax^{2} - a)^{3}} - 8(\frac{-3(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{4}})ax - \frac{8a}{(x + a^{2}x - ax^{2} - a)^{3}} + 2(\frac{-2(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{3}})a + 0 + 2(\frac{-3(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{4}})a^{4} + 0 + 4(\frac{-3(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{4}})a^{2} + 0 + 2(\frac{-3(1 + a^{2} - a*2x + 0)}{(x + a^{2}x - ax^{2} - a)^{4}})\\=&\frac{48a^{3}x^{11}}{(x + a^{2}x - ax^{2} - a)^{4}} + \frac{96a^{3}x^{7}}{(x + a^{2}x - ax^{2} - a)^{4}} + \frac{36ax^{9}}{(x + a^{2}x - ax^{2} - a)^{4}} + \frac{216a^{2}x^{9}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{72a^{2}x^{10}}{(x + a^{2}x - ax^{2} - a)^{4}} - \frac{144a^{4}x^{6}}{(x + a^{2}x - ax^{2} - a)^{4}} + \frac{72ax^{5}}{(x + a^{2}x - ax^{2} - a)^{4}} + \frac{240a^{2}x^{5}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{72a^{4}x^{10}}{(x + a^{2}x - ax^{2} - a)^{4}} - \frac{144a^{2}x^{6}}{(x + a^{2}x - ax^{2} - a)^{4}} - \frac{204ax^{8}}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{384ax^{7}}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{72a^{3}x^{9}}{(x + a^{2}x - ax^{2} - a)^{4}} + \frac{48a^{3}x^{3}}{(x + a^{2}x - ax^{2} - a)^{4}} + \frac{144a^{3}x^{5}}{(x + a^{2}x - ax^{2} - a)^{4}} - \frac{216a^{3}x^{4}}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{36a^{5}x^{9}}{(x + a^{2}x - ax^{2} - a)^{4}} - \frac{204a^{3}x^{8}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{72a^{4}x^{2}}{(x + a^{2}x - ax^{2} - a)^{4}} + \frac{72a^{5}x^{5}}{(x + a^{2}x - ax^{2} - a)^{4}} - \frac{216ax^{4}}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{192ax^{3}}{(x + a^{2}x - ax^{2} - a)^{2}} - \frac{72a^{2}x^{2}}{(x + a^{2}x - ax^{2} - a)^{4}} - \frac{18a^{2}x^{8}}{(x + a^{2}x - ax^{2} - a)^{4}} + \frac{96a^{2}x^{7}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{6x^{8}}{(x + a^{2}x - ax^{2} - a)^{4}} + \frac{48x^{7}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{168x^{6}}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{336x^{5}}{(x + a^{2}x - ax^{2} - a)} + \frac{24a^{2}x}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{36a^{4}x^{4}}{(x + a^{2}x - ax^{2} - a)^{4}} - \frac{36a^{2}x^{4}}{(x + a^{2}x - ax^{2} - a)^{4}} + \frac{96a^{2}x^{3}}{(x + a^{2}x - ax^{2} - a)^{3}} + \frac{48a^{4}x^{3}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{72a^{2}x^{2}}{(x + a^{2}x - ax^{2} - a)^{2}} - \frac{6a^{6}x^{8}}{(x + a^{2}x - ax^{2} - a)^{4}} - \frac{18a^{4}x^{8}}{(x + a^{2}x - ax^{2} - a)^{4}} + \frac{48a^{4}x^{7}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{168a^{2}x^{6}}{(x + a^{2}x - ax^{2} - a)^{2}} - \frac{12x^{4}}{(x + a^{2}x - ax^{2} - a)^{4}} + \frac{48x^{3}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{72x^{2}}{(x + a^{2}x - ax^{2} - a)^{2}} + \frac{48x}{(x + a^{2}x - ax^{2} - a)} + \frac{36a^{5}x}{(x + a^{2}x - ax^{2} - a)^{4}} + \frac{72a^{3}x}{(x + a^{2}x - ax^{2} - a)^{4}} - \frac{12a^{6}x^{4}}{(x + a^{2}x - ax^{2} - a)^{4}} + \frac{36ax}{(x + a^{2}x - ax^{2} - a)^{4}} - \frac{12a^{3}}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{12a}{(x + a^{2}x - ax^{2} - a)^{3}} - \frac{18a^{4}}{(x + a^{2}x - ax^{2} - a)^{4}} - \frac{6a^{6}}{(x + a^{2}x - ax^{2} - a)^{4}} - \frac{18a^{2}}{(x + a^{2}x - ax^{2} - a)^{4}} - \frac{6}{(x + a^{2}x - ax^{2} - a)^{4}}\\ \end{split}\end{equation} \]