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 1 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ ((1 - {a}^{2}){x}^{2} + 2a(1 - a)x + {a}^{2}{t}^{2})({\frac{1}{(x + a)}}^{2} + (({a}^{2} + 1){t}^{2} - 2x{(a - 1)}^{2}){\frac{1}{({(a - 1)}^{2}x + a{t}^{2})}}^{2})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{x^{2}}{(x + a)^{2}} + \frac{t^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{4ax^{3}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{2x^{3}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{a^{2}x^{2}}{(x + a)^{2}} - \frac{a^{4}t^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{2a^{4}x^{3}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{4a^{3}x^{3}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{2ax}{(x + a)^{2}} + \frac{6a^{3}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{2at^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{12a^{3}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{12a^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{4ax^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{2a^{2}x}{(x + a)^{2}} - \frac{4a^{4}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{4a^{2}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{4a^{4}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{a^{2}t^{4}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{a^{4}t^{4}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{a^{2}t^{2}}{(x + a)^{2}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x^{2}}{(x + a)^{2}} + \frac{t^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{4ax^{3}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{2x^{3}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{a^{2}x^{2}}{(x + a)^{2}} - \frac{a^{4}t^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{2a^{4}x^{3}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{4a^{3}x^{3}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{2ax}{(x + a)^{2}} + \frac{6a^{3}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{2at^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{12a^{3}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{12a^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{4ax^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{2a^{2}x}{(x + a)^{2}} - \frac{4a^{4}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{4a^{2}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{4a^{4}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{a^{2}t^{4}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{a^{4}t^{4}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{a^{2}t^{2}}{(x + a)^{2}}\right)}{dx}\\=&(\frac{-2(1 + 0)}{(x + a)^{3}})x^{2} + \frac{2x}{(x + a)^{2}} + (\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})t^{2}x^{2} + \frac{t^{2}*2x}{(a^{2}x - 2ax + x + at^{2})^{2}} + 4(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})ax^{3} + \frac{4a*3x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - 2(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})x^{3} - \frac{2*3x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - (\frac{-2(1 + 0)}{(x + a)^{3}})a^{2}x^{2} - \frac{a^{2}*2x}{(x + a)^{2}} - (\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{4}t^{2}x^{2} - \frac{a^{4}t^{2}*2x}{(a^{2}x - 2ax + x + at^{2})^{2}} + 2(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{4}x^{3} + \frac{2a^{4}*3x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - 4(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{3}x^{3} - \frac{4a^{3}*3x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + 2(\frac{-2(1 + 0)}{(x + a)^{3}})ax + \frac{2a}{(x + a)^{2}} + 6(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{3}t^{2}x + \frac{6a^{3}t^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + 2(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})at^{2}x + \frac{2at^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - 12(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{3}x^{2} - \frac{12a^{3}*2x}{(a^{2}x - 2ax + x + at^{2})^{2}} + 12(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{2}x^{2} + \frac{12a^{2}*2x}{(a^{2}x - 2ax + x + at^{2})^{2}} - 4(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})ax^{2} - \frac{4a*2x}{(a^{2}x - 2ax + x + at^{2})^{2}} - 2(\frac{-2(1 + 0)}{(x + a)^{3}})a^{2}x - \frac{2a^{2}}{(x + a)^{2}} - 4(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{4}t^{2}x - \frac{4a^{4}t^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - 4(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{2}t^{2}x - \frac{4a^{2}t^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + 4(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{4}x^{2} + \frac{4a^{4}*2x}{(a^{2}x - 2ax + x + at^{2})^{2}} + (\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{2}t^{4} + 0 + (\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{4}t^{4} + 0 + (\frac{-2(1 + 0)}{(x + a)^{3}})a^{2}t^{2} + 0\\=&\frac{-2x^{2}}{(x + a)^{3}} + \frac{2x}{(x + a)^{2}} - \frac{2a^{2}t^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{4at^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{2t^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{2t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{20a^{2}x^{3}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{16ax^{3}}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{12ax^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{4x^{3}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{6x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{2a^{2}x^{2}}{(x + a)^{3}} - \frac{2a^{2}x}{(x + a)^{2}} + \frac{2a^{6}t^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{4a^{5}t^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{2a^{4}t^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{2a^{4}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{4a^{6}x^{3}}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{16a^{5}x^{3}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{20a^{4}x^{3}}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{6a^{4}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{12a^{3}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{4ax}{(x + a)^{3}} - \frac{28a^{5}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{40a^{4}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{32a^{3}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{16a^{2}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{4at^{2}x}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{8a^{6}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{2at^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{4a^{4}t^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{40a^{5}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{80a^{4}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{80a^{3}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{24a^{3}x}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{40a^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{24a^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{8ax^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{8ax}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{4a^{2}x}{(x + a)^{3}} - \frac{4a^{2}t^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{6a^{3}t^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{8a^{4}x}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{8a^{6}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{4a^{4}t^{4}}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{4a^{3}t^{4}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{2a^{2}t^{4}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{2a^{6}t^{4}}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{4a^{5}t^{4}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{2a^{2}t^{2}}{(x + a)^{3}} - \frac{2a^{2}}{(x + a)^{2}} + \frac{2a}{(x + a)^{2}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ ((1 - {a}^{2}){x}^{2} + 2a(1 - a)x + {a}^{2}{t}^{2})({\frac{1}{(x + a)}}^{2} + (({a}^{2} + 1){t}^{2} - 2x{(a - 1)}^{2}){\frac{1}{({(a - 1)}^{2}x + a{t}^{2})}}^{2})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{x^{2}}{(x + a)^{2}} + \frac{t^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{4ax^{3}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{2x^{3}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{a^{2}x^{2}}{(x + a)^{2}} - \frac{a^{4}t^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{2a^{4}x^{3}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{4a^{3}x^{3}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{2ax}{(x + a)^{2}} + \frac{6a^{3}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{2at^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{12a^{3}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{12a^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{4ax^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{2a^{2}x}{(x + a)^{2}} - \frac{4a^{4}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{4a^{2}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{4a^{4}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{a^{2}t^{4}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{a^{4}t^{4}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{a^{2}t^{2}}{(x + a)^{2}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x^{2}}{(x + a)^{2}} + \frac{t^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{4ax^{3}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{2x^{3}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{a^{2}x^{2}}{(x + a)^{2}} - \frac{a^{4}t^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{2a^{4}x^{3}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{4a^{3}x^{3}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{2ax}{(x + a)^{2}} + \frac{6a^{3}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{2at^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{12a^{3}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{12a^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{4ax^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{2a^{2}x}{(x + a)^{2}} - \frac{4a^{4}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{4a^{2}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{4a^{4}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{a^{2}t^{4}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{a^{4}t^{4}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{a^{2}t^{2}}{(x + a)^{2}}\right)}{dx}\\=&(\frac{-2(1 + 0)}{(x + a)^{3}})x^{2} + \frac{2x}{(x + a)^{2}} + (\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})t^{2}x^{2} + \frac{t^{2}*2x}{(a^{2}x - 2ax + x + at^{2})^{2}} + 4(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})ax^{3} + \frac{4a*3x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - 2(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})x^{3} - \frac{2*3x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - (\frac{-2(1 + 0)}{(x + a)^{3}})a^{2}x^{2} - \frac{a^{2}*2x}{(x + a)^{2}} - (\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{4}t^{2}x^{2} - \frac{a^{4}t^{2}*2x}{(a^{2}x - 2ax + x + at^{2})^{2}} + 2(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{4}x^{3} + \frac{2a^{4}*3x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - 4(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{3}x^{3} - \frac{4a^{3}*3x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + 2(\frac{-2(1 + 0)}{(x + a)^{3}})ax + \frac{2a}{(x + a)^{2}} + 6(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{3}t^{2}x + \frac{6a^{3}t^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + 2(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})at^{2}x + \frac{2at^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - 12(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{3}x^{2} - \frac{12a^{3}*2x}{(a^{2}x - 2ax + x + at^{2})^{2}} + 12(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{2}x^{2} + \frac{12a^{2}*2x}{(a^{2}x - 2ax + x + at^{2})^{2}} - 4(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})ax^{2} - \frac{4a*2x}{(a^{2}x - 2ax + x + at^{2})^{2}} - 2(\frac{-2(1 + 0)}{(x + a)^{3}})a^{2}x - \frac{2a^{2}}{(x + a)^{2}} - 4(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{4}t^{2}x - \frac{4a^{4}t^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - 4(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{2}t^{2}x - \frac{4a^{2}t^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + 4(\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{4}x^{2} + \frac{4a^{4}*2x}{(a^{2}x - 2ax + x + at^{2})^{2}} + (\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{2}t^{4} + 0 + (\frac{-2(a^{2} - 2a + 1 + 0)}{(a^{2}x - 2ax + x + at^{2})^{3}})a^{4}t^{4} + 0 + (\frac{-2(1 + 0)}{(x + a)^{3}})a^{2}t^{2} + 0\\=&\frac{-2x^{2}}{(x + a)^{3}} + \frac{2x}{(x + a)^{2}} - \frac{2a^{2}t^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{4at^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{2t^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{2t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{20a^{2}x^{3}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{16ax^{3}}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{12ax^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{4x^{3}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{6x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{2a^{2}x^{2}}{(x + a)^{3}} - \frac{2a^{2}x}{(x + a)^{2}} + \frac{2a^{6}t^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{4a^{5}t^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{2a^{4}t^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{2a^{4}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{4a^{6}x^{3}}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{16a^{5}x^{3}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{20a^{4}x^{3}}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{6a^{4}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{12a^{3}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{4ax}{(x + a)^{3}} - \frac{28a^{5}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{40a^{4}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{32a^{3}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{16a^{2}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{4at^{2}x}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{8a^{6}t^{2}x}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{2at^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{4a^{4}t^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{40a^{5}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{80a^{4}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{80a^{3}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{24a^{3}x}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{40a^{2}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{24a^{2}x}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{8ax^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{8ax}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{4a^{2}x}{(x + a)^{3}} - \frac{4a^{2}t^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{6a^{3}t^{2}}{(a^{2}x - 2ax + x + at^{2})^{2}} + \frac{8a^{4}x}{(a^{2}x - 2ax + x + at^{2})^{2}} - \frac{8a^{6}x^{2}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{4a^{4}t^{4}}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{4a^{3}t^{4}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{2a^{2}t^{4}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{2a^{6}t^{4}}{(a^{2}x - 2ax + x + at^{2})^{3}} + \frac{4a^{5}t^{4}}{(a^{2}x - 2ax + x + at^{2})^{3}} - \frac{2a^{2}t^{2}}{(x + a)^{3}} - \frac{2a^{2}}{(x + a)^{2}} + \frac{2a}{(x + a)^{2}}\\ \end{split}\end{equation} \]