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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{x}{(4 - xx - sqrt(4 - xx))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)}\right)}{dx}\\=&(\frac{-(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}})x + \frac{1}{(-x^{2} - sqrt(-x^{2} + 4) + 4)}\\=&\frac{2x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{1}{(-x^{2} - sqrt(-x^{2} + 4) + 4)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{2x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{1}{(-x^{2} - sqrt(-x^{2} + 4) + 4)}\right)}{dx}\\=&2(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})x^{2} + \frac{2*2x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})x^{2}}{(-x^{2} + 4)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{2x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{1}{2}}} + (\frac{-(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}})\\=&\frac{8x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{8x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{6x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} + \frac{2x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)} - \frac{x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{3}{2}}} - \frac{2x}{(-x^{2} + 4)^{\frac{1}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{8x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{8x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{6x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} + \frac{2x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)} - \frac{x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{3}{2}}} - \frac{2x}{(-x^{2} + 4)^{\frac{1}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{1}{2}}}\right)}{dx}\\=&8(\frac{-3(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}})x^{3} + \frac{8*3x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{8(\frac{-3(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}})x^{3}}{(-x^{2} + 4)^{\frac{1}{2}}} - \frac{8(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{8*3x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{1}{2}}} + 6(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})x + \frac{6}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} + \frac{2(\frac{-3(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}})x^{3}}{(-x^{2} + 4)} + \frac{2(\frac{-(-2x + 0)}{(-x^{2} + 4)^{2}})x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} + \frac{2*3x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)} - \frac{(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})x^{3}}{(-x^{2} + 4)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{5}{2}}})x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{3x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{3}{2}}} - \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{2(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})x}{(-x^{2} + 4)^{\frac{1}{2}}} - \frac{2}{(-x^{2} + 4)^{\frac{1}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})x}{(-x^{2} + 4)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{1}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{1}{2}}}\\=&\frac{48x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}} - \frac{72x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{48x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} + \frac{36x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)} - \frac{12x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{3}{2}}} - \frac{24x^{2}}{(-x^{2} + 4)^{\frac{1}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{24x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{6x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{2}} - \frac{6x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{\frac{3}{2}}} + \frac{6x^{2}}{(-x^{2} + 4)(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{3x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{5}{2}}} - \frac{5x^{2}}{(-x^{2} + 4)^{\frac{3}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{3}{2}}} + \frac{6x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)} - \frac{1}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{1}{2}}} - \frac{2}{(-x^{2} + 4)^{\frac{1}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} + \frac{6}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{48x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}} - \frac{72x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{48x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} + \frac{36x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)} - \frac{12x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{3}{2}}} - \frac{24x^{2}}{(-x^{2} + 4)^{\frac{1}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{24x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{6x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{2}} - \frac{6x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{\frac{3}{2}}} + \frac{6x^{2}}{(-x^{2} + 4)(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{3x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{5}{2}}} - \frac{5x^{2}}{(-x^{2} + 4)^{\frac{3}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{3}{2}}} + \frac{6x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)} - \frac{1}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{1}{2}}} - \frac{2}{(-x^{2} + 4)^{\frac{1}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} + \frac{6}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}}\right)}{dx}\\=&48(\frac{-4(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{5}})x^{4} + \frac{48*4x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}} - \frac{72(\frac{-4(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{5}})x^{4}}{(-x^{2} + 4)^{\frac{1}{2}}} - \frac{72(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}} - \frac{72*4x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{\frac{1}{2}}} + 48(\frac{-3(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}})x^{2} + \frac{48*2x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} + \frac{36(\frac{-4(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{5}})x^{4}}{(-x^{2} + 4)} + \frac{36(\frac{-(-2x + 0)}{(-x^{2} + 4)^{2}})x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}} + \frac{36*4x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)} - \frac{12(\frac{-3(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}})x^{4}}{(-x^{2} + 4)^{\frac{3}{2}}} - \frac{12(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{5}{2}}})x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{12*4x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{3}{2}}} - \frac{24(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{24(\frac{-3(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}})x^{2}}{(-x^{2} + 4)^{\frac{1}{2}}} - \frac{24*2x}{(-x^{2} + 4)^{\frac{1}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{24(\frac{-3(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}})x^{2}}{(-x^{2} + 4)^{\frac{1}{2}}} - \frac{24(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{24*2x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{6(\frac{-3(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}})x^{4}}{(-x^{2} + 4)^{2}} + \frac{6(\frac{-2(-2x + 0)}{(-x^{2} + 4)^{3}})x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} + \frac{6*4x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{2}} - \frac{6(\frac{-4(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{5}})x^{4}}{(-x^{2} + 4)^{\frac{3}{2}}} - \frac{6(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{5}{2}}})x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}} - \frac{6*4x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{\frac{3}{2}}} + \frac{6(\frac{-(-2x + 0)}{(-x^{2} + 4)^{2}})x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} + \frac{6(\frac{-3(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}})x^{2}}{(-x^{2} + 4)} + \frac{6*2x}{(-x^{2} + 4)(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{3(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})x^{4}}{(-x^{2} + 4)^{\frac{5}{2}}} - \frac{3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{7}{2}}})x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{3*4x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{5}{2}}} - \frac{5(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{5}{2}}})x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{5(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})x^{2}}{(-x^{2} + 4)^{\frac{3}{2}}} - \frac{5*2x}{(-x^{2} + 4)^{\frac{3}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})x^{2}}{(-x^{2} + 4)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{5}{2}}})x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{2x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{3}{2}}} + \frac{6(\frac{-3(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}})x^{2}}{(-x^{2} + 4)} + \frac{6(\frac{-(-2x + 0)}{(-x^{2} + 4)^{2}})x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} + \frac{6*2x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)} - \frac{(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})}{(-x^{2} + 4)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{2(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})}{(-x^{2} + 4)^{\frac{1}{2}}} + 6(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})\\=&\frac{384x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{5}} - \frac{768x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{5}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{480x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}} + \frac{576x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{5}(-x^{2} + 4)} - \frac{144x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{\frac{3}{2}}} - \frac{288x^{3}}{(-x^{2} + 4)^{\frac{1}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}} - \frac{432x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{120x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{192x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{5}(-x^{2} + 4)^{\frac{3}{2}}} + \frac{144x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{2}} + \frac{144x^{3}}{(-x^{2} + 4)(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}} - \frac{48x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{5}{2}}} - \frac{72x^{3}}{(-x^{2} + 4)^{\frac{3}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} + \frac{216x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)} - \frac{72x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{1}{2}}} - \frac{48x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{3}{2}}} - \frac{48x}{(-x^{2} + 4)^{\frac{1}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{36x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{\frac{5}{2}}} + \frac{30x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{3}} + \frac{36x^{3}}{(-x^{2} + 4)^{2}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} + \frac{24x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{5}(-x^{2} + 4)^{2}} - \frac{24x^{3}}{(-x^{2} + 4)^{\frac{3}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}} - \frac{36x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{\frac{3}{2}}} + \frac{18x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)} - \frac{15x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{7}{2}}} - \frac{27x^{3}}{(-x^{2} + 4)^{\frac{5}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} + \frac{24x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{2}} - \frac{11x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{3}{2}}} - \frac{3x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{5}{2}}} - \frac{4x}{(-x^{2} + 4)^{\frac{3}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} + \frac{12x}{(-x^{2} + 4)(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{x}{(4 - xx - sqrt(4 - xx))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)}\right)}{dx}\\=&(\frac{-(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}})x + \frac{1}{(-x^{2} - sqrt(-x^{2} + 4) + 4)}\\=&\frac{2x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{1}{(-x^{2} - sqrt(-x^{2} + 4) + 4)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{2x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{1}{(-x^{2} - sqrt(-x^{2} + 4) + 4)}\right)}{dx}\\=&2(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})x^{2} + \frac{2*2x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})x^{2}}{(-x^{2} + 4)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{2x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{1}{2}}} + (\frac{-(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}})\\=&\frac{8x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{8x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{6x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} + \frac{2x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)} - \frac{x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{3}{2}}} - \frac{2x}{(-x^{2} + 4)^{\frac{1}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{8x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{8x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{6x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} + \frac{2x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)} - \frac{x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{3}{2}}} - \frac{2x}{(-x^{2} + 4)^{\frac{1}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{1}{2}}}\right)}{dx}\\=&8(\frac{-3(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}})x^{3} + \frac{8*3x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{8(\frac{-3(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}})x^{3}}{(-x^{2} + 4)^{\frac{1}{2}}} - \frac{8(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{8*3x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{1}{2}}} + 6(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})x + \frac{6}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} + \frac{2(\frac{-3(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}})x^{3}}{(-x^{2} + 4)} + \frac{2(\frac{-(-2x + 0)}{(-x^{2} + 4)^{2}})x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} + \frac{2*3x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)} - \frac{(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})x^{3}}{(-x^{2} + 4)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{5}{2}}})x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{3x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{3}{2}}} - \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{2(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})x}{(-x^{2} + 4)^{\frac{1}{2}}} - \frac{2}{(-x^{2} + 4)^{\frac{1}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})x}{(-x^{2} + 4)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{1}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{1}{2}}}\\=&\frac{48x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}} - \frac{72x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{48x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} + \frac{36x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)} - \frac{12x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{3}{2}}} - \frac{24x^{2}}{(-x^{2} + 4)^{\frac{1}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{24x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{6x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{2}} - \frac{6x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{\frac{3}{2}}} + \frac{6x^{2}}{(-x^{2} + 4)(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{3x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{5}{2}}} - \frac{5x^{2}}{(-x^{2} + 4)^{\frac{3}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{3}{2}}} + \frac{6x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)} - \frac{1}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{1}{2}}} - \frac{2}{(-x^{2} + 4)^{\frac{1}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} + \frac{6}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{48x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}} - \frac{72x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{48x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} + \frac{36x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)} - \frac{12x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{3}{2}}} - \frac{24x^{2}}{(-x^{2} + 4)^{\frac{1}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{24x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{6x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{2}} - \frac{6x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{\frac{3}{2}}} + \frac{6x^{2}}{(-x^{2} + 4)(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{3x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{5}{2}}} - \frac{5x^{2}}{(-x^{2} + 4)^{\frac{3}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{3}{2}}} + \frac{6x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)} - \frac{1}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{1}{2}}} - \frac{2}{(-x^{2} + 4)^{\frac{1}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} + \frac{6}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}}\right)}{dx}\\=&48(\frac{-4(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{5}})x^{4} + \frac{48*4x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}} - \frac{72(\frac{-4(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{5}})x^{4}}{(-x^{2} + 4)^{\frac{1}{2}}} - \frac{72(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}} - \frac{72*4x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{\frac{1}{2}}} + 48(\frac{-3(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}})x^{2} + \frac{48*2x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} + \frac{36(\frac{-4(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{5}})x^{4}}{(-x^{2} + 4)} + \frac{36(\frac{-(-2x + 0)}{(-x^{2} + 4)^{2}})x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}} + \frac{36*4x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)} - \frac{12(\frac{-3(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}})x^{4}}{(-x^{2} + 4)^{\frac{3}{2}}} - \frac{12(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{5}{2}}})x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{12*4x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{3}{2}}} - \frac{24(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{24(\frac{-3(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}})x^{2}}{(-x^{2} + 4)^{\frac{1}{2}}} - \frac{24*2x}{(-x^{2} + 4)^{\frac{1}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{24(\frac{-3(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}})x^{2}}{(-x^{2} + 4)^{\frac{1}{2}}} - \frac{24(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{24*2x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{6(\frac{-3(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}})x^{4}}{(-x^{2} + 4)^{2}} + \frac{6(\frac{-2(-2x + 0)}{(-x^{2} + 4)^{3}})x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} + \frac{6*4x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{2}} - \frac{6(\frac{-4(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{5}})x^{4}}{(-x^{2} + 4)^{\frac{3}{2}}} - \frac{6(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{5}{2}}})x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}} - \frac{6*4x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{\frac{3}{2}}} + \frac{6(\frac{-(-2x + 0)}{(-x^{2} + 4)^{2}})x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} + \frac{6(\frac{-3(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}})x^{2}}{(-x^{2} + 4)} + \frac{6*2x}{(-x^{2} + 4)(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{3(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})x^{4}}{(-x^{2} + 4)^{\frac{5}{2}}} - \frac{3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{7}{2}}})x^{4}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{3*4x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{5}{2}}} - \frac{5(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{5}{2}}})x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{5(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})x^{2}}{(-x^{2} + 4)^{\frac{3}{2}}} - \frac{5*2x}{(-x^{2} + 4)^{\frac{3}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})x^{2}}{(-x^{2} + 4)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{5}{2}}})x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{2x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{3}{2}}} + \frac{6(\frac{-3(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}})x^{2}}{(-x^{2} + 4)} + \frac{6(\frac{-(-2x + 0)}{(-x^{2} + 4)^{2}})x^{2}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} + \frac{6*2x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)} - \frac{(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})}{(-x^{2} + 4)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} - \frac{2(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})}{(-x^{2} + 4)^{\frac{1}{2}}} + 6(\frac{-2(-2x - \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 0)}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}})\\=&\frac{384x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{5}} - \frac{768x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{5}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{480x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}} + \frac{576x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{5}(-x^{2} + 4)} - \frac{144x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{\frac{3}{2}}} - \frac{288x^{3}}{(-x^{2} + 4)^{\frac{1}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}} - \frac{432x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{\frac{1}{2}}} + \frac{120x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{192x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{5}(-x^{2} + 4)^{\frac{3}{2}}} + \frac{144x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{2}} + \frac{144x^{3}}{(-x^{2} + 4)(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}} - \frac{48x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{5}{2}}} - \frac{72x^{3}}{(-x^{2} + 4)^{\frac{3}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} + \frac{216x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)} - \frac{72x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{1}{2}}} - \frac{48x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{\frac{3}{2}}} - \frac{48x}{(-x^{2} + 4)^{\frac{1}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} - \frac{36x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{\frac{5}{2}}} + \frac{30x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{3}} + \frac{36x^{3}}{(-x^{2} + 4)^{2}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}} + \frac{24x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{5}(-x^{2} + 4)^{2}} - \frac{24x^{3}}{(-x^{2} + 4)^{\frac{3}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}} - \frac{36x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{4}(-x^{2} + 4)^{\frac{3}{2}}} + \frac{18x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)} - \frac{15x^{5}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{7}{2}}} - \frac{27x^{3}}{(-x^{2} + 4)^{\frac{5}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} + \frac{24x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}(-x^{2} + 4)^{2}} - \frac{11x}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{3}{2}}} - \frac{3x^{3}}{(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}(-x^{2} + 4)^{\frac{5}{2}}} - \frac{4x}{(-x^{2} + 4)^{\frac{3}{2}}(-x^{2} - sqrt(-x^{2} + 4) + 4)^{2}} + \frac{12x}{(-x^{2} + 4)(-x^{2} - sqrt(-x^{2} + 4) + 4)^{3}}\\ \end{split}\end{equation} \]