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 4 questions in this calculation: for each question, the 4 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/4]Find\ the\ 4th\ derivative\ of\ function\ \frac{x}{}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( 0\right)}{dx}\\=&0\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{dx}\\=&0\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{dx}\\=&0\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{dx}\\=&0\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/4]Find\ the\ 4th\ derivative\ of\ function\ sqrt(-{x}^{2} - 1)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = sqrt(-x^{2} - 1)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sqrt(-x^{2} - 1)\right)}{dx}\\=&\frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} - 1)^{\frac{1}{2}}}\\=&\frac{-x}{(-x^{2} - 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-x}{(-x^{2} - 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} - 1)^{\frac{3}{2}}})x - \frac{1}{(-x^{2} - 1)^{\frac{1}{2}}}\\=&\frac{-x^{2}}{(-x^{2} - 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} - 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-x^{2}}{(-x^{2} - 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} - 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} - 1)^{\frac{5}{2}}})x^{2} - \frac{2x}{(-x^{2} - 1)^{\frac{3}{2}}} - (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} - 1)^{\frac{3}{2}}})\\=&\frac{-3x^{3}}{(-x^{2} - 1)^{\frac{5}{2}}} - \frac{3x}{(-x^{2} - 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-3x^{3}}{(-x^{2} - 1)^{\frac{5}{2}}} - \frac{3x}{(-x^{2} - 1)^{\frac{3}{2}}}\right)}{dx}\\=&-3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} - 1)^{\frac{7}{2}}})x^{3} - \frac{3*3x^{2}}{(-x^{2} - 1)^{\frac{5}{2}}} - 3(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} - 1)^{\frac{5}{2}}})x - \frac{3}{(-x^{2} - 1)^{\frac{3}{2}}}\\=&\frac{-15x^{4}}{(-x^{2} - 1)^{\frac{7}{2}}} - \frac{18x^{2}}{(-x^{2} - 1)^{\frac{5}{2}}} - \frac{3}{(-x^{2} - 1)^{\frac{3}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/4]Find\ the\ 4th\ derivative\ of\ function\ lg(-1 - e^{x})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = lg(-e^{x} - 1)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg(-e^{x} - 1)\right)}{dx}\\=&\frac{(-e^{x} + 0)}{ln{10}(-e^{x} - 1)}\\=&\frac{-e^{x}}{(-e^{x} - 1)ln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-e^{x}}{(-e^{x} - 1)ln{10}}\right)}{dx}\\=&\frac{-(\frac{-(-e^{x} + 0)}{(-e^{x} - 1)^{2}})e^{x}}{ln{10}} - \frac{e^{x}}{(-e^{x} - 1)ln{10}} - \frac{e^{x}*-0}{(-e^{x} - 1)ln^{2}{10}}\\=&\frac{-e^{{x}*{2}}}{(-e^{x} - 1)^{2}ln{10}} - \frac{e^{x}}{(-e^{x} - 1)ln{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-e^{{x}*{2}}}{(-e^{x} - 1)^{2}ln{10}} - \frac{e^{x}}{(-e^{x} - 1)ln{10}}\right)}{dx}\\=&\frac{-(\frac{-2(-e^{x} + 0)}{(-e^{x} - 1)^{3}})e^{{x}*{2}}}{ln{10}} - \frac{2e^{x}e^{x}}{(-e^{x} - 1)^{2}ln{10}} - \frac{e^{{x}*{2}}*-0}{(-e^{x} - 1)^{2}ln^{2}{10}} - \frac{(\frac{-(-e^{x} + 0)}{(-e^{x} - 1)^{2}})e^{x}}{ln{10}} - \frac{e^{x}}{(-e^{x} - 1)ln{10}} - \frac{e^{x}*-0}{(-e^{x} - 1)ln^{2}{10}}\\=&\frac{-2e^{{x}*{3}}}{(-e^{x} - 1)^{3}ln{10}} - \frac{3e^{{x}*{2}}}{(-e^{x} - 1)^{2}ln{10}} - \frac{e^{x}}{(-e^{x} - 1)ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2e^{{x}*{3}}}{(-e^{x} - 1)^{3}ln{10}} - \frac{3e^{{x}*{2}}}{(-e^{x} - 1)^{2}ln{10}} - \frac{e^{x}}{(-e^{x} - 1)ln{10}}\right)}{dx}\\=&\frac{-2(\frac{-3(-e^{x} + 0)}{(-e^{x} - 1)^{4}})e^{{x}*{3}}}{ln{10}} - \frac{2*3e^{{x}*{2}}e^{x}}{(-e^{x} - 1)^{3}ln{10}} - \frac{2e^{{x}*{3}}*-0}{(-e^{x} - 1)^{3}ln^{2}{10}} - \frac{3(\frac{-2(-e^{x} + 0)}{(-e^{x} - 1)^{3}})e^{{x}*{2}}}{ln{10}} - \frac{3*2e^{x}e^{x}}{(-e^{x} - 1)^{2}ln{10}} - \frac{3e^{{x}*{2}}*-0}{(-e^{x} - 1)^{2}ln^{2}{10}} - \frac{(\frac{-(-e^{x} + 0)}{(-e^{x} - 1)^{2}})e^{x}}{ln{10}} - \frac{e^{x}}{(-e^{x} - 1)ln{10}} - \frac{e^{x}*-0}{(-e^{x} - 1)ln^{2}{10}}\\=&\frac{-6e^{{x}*{4}}}{(-e^{x} - 1)^{4}ln{10}} - \frac{12e^{{x}*{3}}}{(-e^{x} - 1)^{3}ln{10}} - \frac{7e^{{x}*{2}}}{(-e^{x} - 1)^{2}ln{10}} - \frac{e^{x}}{(-e^{x} - 1)ln{10}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[4/4]Find\ the\ 4th\ derivative\ of\ function\ arccos({x}^{2} + 2)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = arccos(x^{2} + 2)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( arccos(x^{2} + 2)\right)}{dx}\\=&(\frac{-(2x + 0)}{((1 - (x^{2} + 2)^{2})^{\frac{1}{2}})})\\=&\frac{-2x}{(-x^{4} - 4x^{2} - 3)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2x}{(-x^{4} - 4x^{2} - 3)^{\frac{1}{2}}}\right)}{dx}\\=&-2(\frac{\frac{-1}{2}(-4x^{3} - 4*2x + 0)}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}})x - \frac{2}{(-x^{4} - 4x^{2} - 3)^{\frac{1}{2}}}\\=&\frac{-4x^{4}}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}} - \frac{8x^{2}}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}} - \frac{2}{(-x^{4} - 4x^{2} - 3)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-4x^{4}}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}} - \frac{8x^{2}}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}} - \frac{2}{(-x^{4} - 4x^{2} - 3)^{\frac{1}{2}}}\right)}{dx}\\=&-4(\frac{\frac{-3}{2}(-4x^{3} - 4*2x + 0)}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}})x^{4} - \frac{4*4x^{3}}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}} - 8(\frac{\frac{-3}{2}(-4x^{3} - 4*2x + 0)}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}})x^{2} - \frac{8*2x}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}} - 2(\frac{\frac{-1}{2}(-4x^{3} - 4*2x + 0)}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}})\\=&\frac{-24x^{7}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - \frac{96x^{5}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - \frac{20x^{3}}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}} - \frac{96x^{3}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - \frac{24x}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-24x^{7}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - \frac{96x^{5}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - \frac{20x^{3}}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}} - \frac{96x^{3}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - \frac{24x}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}}\right)}{dx}\\=&-24(\frac{\frac{-5}{2}(-4x^{3} - 4*2x + 0)}{(-x^{4} - 4x^{2} - 3)^{\frac{7}{2}}})x^{7} - \frac{24*7x^{6}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - 96(\frac{\frac{-5}{2}(-4x^{3} - 4*2x + 0)}{(-x^{4} - 4x^{2} - 3)^{\frac{7}{2}}})x^{5} - \frac{96*5x^{4}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - 20(\frac{\frac{-3}{2}(-4x^{3} - 4*2x + 0)}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}})x^{3} - \frac{20*3x^{2}}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}} - 96(\frac{\frac{-5}{2}(-4x^{3} - 4*2x + 0)}{(-x^{4} - 4x^{2} - 3)^{\frac{7}{2}}})x^{3} - \frac{96*3x^{2}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - 24(\frac{\frac{-3}{2}(-4x^{3} - 4*2x + 0)}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}})x - \frac{24}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}}\\=&\frac{-240x^{10}}{(-x^{4} - 4x^{2} - 3)^{\frac{7}{2}}} - \frac{1440x^{8}}{(-x^{4} - 4x^{2} - 3)^{\frac{7}{2}}} - \frac{288x^{6}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - \frac{2880x^{6}}{(-x^{4} - 4x^{2} - 3)^{\frac{7}{2}}} - \frac{864x^{4}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - \frac{60x^{2}}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}} - \frac{1920x^{4}}{(-x^{4} - 4x^{2} - 3)^{\frac{7}{2}}} - \frac{576x^{2}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - \frac{24}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/4]Find\ the\ 4th\ derivative\ of\ function\ \frac{x}{}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( 0\right)}{dx}\\=&0\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{dx}\\=&0\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{dx}\\=&0\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 0\right)}{dx}\\=&0\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/4]Find\ the\ 4th\ derivative\ of\ function\ sqrt(-{x}^{2} - 1)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = sqrt(-x^{2} - 1)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sqrt(-x^{2} - 1)\right)}{dx}\\=&\frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} - 1)^{\frac{1}{2}}}\\=&\frac{-x}{(-x^{2} - 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-x}{(-x^{2} - 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} - 1)^{\frac{3}{2}}})x - \frac{1}{(-x^{2} - 1)^{\frac{1}{2}}}\\=&\frac{-x^{2}}{(-x^{2} - 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} - 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-x^{2}}{(-x^{2} - 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} - 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} - 1)^{\frac{5}{2}}})x^{2} - \frac{2x}{(-x^{2} - 1)^{\frac{3}{2}}} - (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} - 1)^{\frac{3}{2}}})\\=&\frac{-3x^{3}}{(-x^{2} - 1)^{\frac{5}{2}}} - \frac{3x}{(-x^{2} - 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-3x^{3}}{(-x^{2} - 1)^{\frac{5}{2}}} - \frac{3x}{(-x^{2} - 1)^{\frac{3}{2}}}\right)}{dx}\\=&-3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} - 1)^{\frac{7}{2}}})x^{3} - \frac{3*3x^{2}}{(-x^{2} - 1)^{\frac{5}{2}}} - 3(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} - 1)^{\frac{5}{2}}})x - \frac{3}{(-x^{2} - 1)^{\frac{3}{2}}}\\=&\frac{-15x^{4}}{(-x^{2} - 1)^{\frac{7}{2}}} - \frac{18x^{2}}{(-x^{2} - 1)^{\frac{5}{2}}} - \frac{3}{(-x^{2} - 1)^{\frac{3}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/4]Find\ the\ 4th\ derivative\ of\ function\ lg(-1 - e^{x})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = lg(-e^{x} - 1)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg(-e^{x} - 1)\right)}{dx}\\=&\frac{(-e^{x} + 0)}{ln{10}(-e^{x} - 1)}\\=&\frac{-e^{x}}{(-e^{x} - 1)ln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-e^{x}}{(-e^{x} - 1)ln{10}}\right)}{dx}\\=&\frac{-(\frac{-(-e^{x} + 0)}{(-e^{x} - 1)^{2}})e^{x}}{ln{10}} - \frac{e^{x}}{(-e^{x} - 1)ln{10}} - \frac{e^{x}*-0}{(-e^{x} - 1)ln^{2}{10}}\\=&\frac{-e^{{x}*{2}}}{(-e^{x} - 1)^{2}ln{10}} - \frac{e^{x}}{(-e^{x} - 1)ln{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-e^{{x}*{2}}}{(-e^{x} - 1)^{2}ln{10}} - \frac{e^{x}}{(-e^{x} - 1)ln{10}}\right)}{dx}\\=&\frac{-(\frac{-2(-e^{x} + 0)}{(-e^{x} - 1)^{3}})e^{{x}*{2}}}{ln{10}} - \frac{2e^{x}e^{x}}{(-e^{x} - 1)^{2}ln{10}} - \frac{e^{{x}*{2}}*-0}{(-e^{x} - 1)^{2}ln^{2}{10}} - \frac{(\frac{-(-e^{x} + 0)}{(-e^{x} - 1)^{2}})e^{x}}{ln{10}} - \frac{e^{x}}{(-e^{x} - 1)ln{10}} - \frac{e^{x}*-0}{(-e^{x} - 1)ln^{2}{10}}\\=&\frac{-2e^{{x}*{3}}}{(-e^{x} - 1)^{3}ln{10}} - \frac{3e^{{x}*{2}}}{(-e^{x} - 1)^{2}ln{10}} - \frac{e^{x}}{(-e^{x} - 1)ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2e^{{x}*{3}}}{(-e^{x} - 1)^{3}ln{10}} - \frac{3e^{{x}*{2}}}{(-e^{x} - 1)^{2}ln{10}} - \frac{e^{x}}{(-e^{x} - 1)ln{10}}\right)}{dx}\\=&\frac{-2(\frac{-3(-e^{x} + 0)}{(-e^{x} - 1)^{4}})e^{{x}*{3}}}{ln{10}} - \frac{2*3e^{{x}*{2}}e^{x}}{(-e^{x} - 1)^{3}ln{10}} - \frac{2e^{{x}*{3}}*-0}{(-e^{x} - 1)^{3}ln^{2}{10}} - \frac{3(\frac{-2(-e^{x} + 0)}{(-e^{x} - 1)^{3}})e^{{x}*{2}}}{ln{10}} - \frac{3*2e^{x}e^{x}}{(-e^{x} - 1)^{2}ln{10}} - \frac{3e^{{x}*{2}}*-0}{(-e^{x} - 1)^{2}ln^{2}{10}} - \frac{(\frac{-(-e^{x} + 0)}{(-e^{x} - 1)^{2}})e^{x}}{ln{10}} - \frac{e^{x}}{(-e^{x} - 1)ln{10}} - \frac{e^{x}*-0}{(-e^{x} - 1)ln^{2}{10}}\\=&\frac{-6e^{{x}*{4}}}{(-e^{x} - 1)^{4}ln{10}} - \frac{12e^{{x}*{3}}}{(-e^{x} - 1)^{3}ln{10}} - \frac{7e^{{x}*{2}}}{(-e^{x} - 1)^{2}ln{10}} - \frac{e^{x}}{(-e^{x} - 1)ln{10}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[4/4]Find\ the\ 4th\ derivative\ of\ function\ arccos({x}^{2} + 2)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = arccos(x^{2} + 2)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( arccos(x^{2} + 2)\right)}{dx}\\=&(\frac{-(2x + 0)}{((1 - (x^{2} + 2)^{2})^{\frac{1}{2}})})\\=&\frac{-2x}{(-x^{4} - 4x^{2} - 3)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2x}{(-x^{4} - 4x^{2} - 3)^{\frac{1}{2}}}\right)}{dx}\\=&-2(\frac{\frac{-1}{2}(-4x^{3} - 4*2x + 0)}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}})x - \frac{2}{(-x^{4} - 4x^{2} - 3)^{\frac{1}{2}}}\\=&\frac{-4x^{4}}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}} - \frac{8x^{2}}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}} - \frac{2}{(-x^{4} - 4x^{2} - 3)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-4x^{4}}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}} - \frac{8x^{2}}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}} - \frac{2}{(-x^{4} - 4x^{2} - 3)^{\frac{1}{2}}}\right)}{dx}\\=&-4(\frac{\frac{-3}{2}(-4x^{3} - 4*2x + 0)}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}})x^{4} - \frac{4*4x^{3}}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}} - 8(\frac{\frac{-3}{2}(-4x^{3} - 4*2x + 0)}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}})x^{2} - \frac{8*2x}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}} - 2(\frac{\frac{-1}{2}(-4x^{3} - 4*2x + 0)}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}})\\=&\frac{-24x^{7}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - \frac{96x^{5}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - \frac{20x^{3}}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}} - \frac{96x^{3}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - \frac{24x}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-24x^{7}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - \frac{96x^{5}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - \frac{20x^{3}}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}} - \frac{96x^{3}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - \frac{24x}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}}\right)}{dx}\\=&-24(\frac{\frac{-5}{2}(-4x^{3} - 4*2x + 0)}{(-x^{4} - 4x^{2} - 3)^{\frac{7}{2}}})x^{7} - \frac{24*7x^{6}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - 96(\frac{\frac{-5}{2}(-4x^{3} - 4*2x + 0)}{(-x^{4} - 4x^{2} - 3)^{\frac{7}{2}}})x^{5} - \frac{96*5x^{4}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - 20(\frac{\frac{-3}{2}(-4x^{3} - 4*2x + 0)}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}})x^{3} - \frac{20*3x^{2}}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}} - 96(\frac{\frac{-5}{2}(-4x^{3} - 4*2x + 0)}{(-x^{4} - 4x^{2} - 3)^{\frac{7}{2}}})x^{3} - \frac{96*3x^{2}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - 24(\frac{\frac{-3}{2}(-4x^{3} - 4*2x + 0)}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}})x - \frac{24}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}}\\=&\frac{-240x^{10}}{(-x^{4} - 4x^{2} - 3)^{\frac{7}{2}}} - \frac{1440x^{8}}{(-x^{4} - 4x^{2} - 3)^{\frac{7}{2}}} - \frac{288x^{6}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - \frac{2880x^{6}}{(-x^{4} - 4x^{2} - 3)^{\frac{7}{2}}} - \frac{864x^{4}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - \frac{60x^{2}}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}} - \frac{1920x^{4}}{(-x^{4} - 4x^{2} - 3)^{\frac{7}{2}}} - \frac{576x^{2}}{(-x^{4} - 4x^{2} - 3)^{\frac{5}{2}}} - \frac{24}{(-x^{4} - 4x^{2} - 3)^{\frac{3}{2}}}\\ \end{split}\end{equation} \]