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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {(1 + x)}^{(\frac{-1}{2})} + {(1 + 8x)}^{(\frac{-1}{2})} + {(\frac{8x}{(8x + 8)})}^{\frac{1}{2}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{1}{(x + 1)^{\frac{1}{2}}} + \frac{1}{(8x + 1)^{\frac{1}{2}}} + \frac{8^{\frac{1}{2}}x^{\frac{1}{2}}}{(8x + 8)^{\frac{1}{2}}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{1}{(x + 1)^{\frac{1}{2}}} + \frac{1}{(8x + 1)^{\frac{1}{2}}} + \frac{8^{\frac{1}{2}}x^{\frac{1}{2}}}{(8x + 8)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(1 + 0)}{(x + 1)^{\frac{3}{2}}}) + (\frac{\frac{-1}{2}(8 + 0)}{(8x + 1)^{\frac{3}{2}}}) + 8^{\frac{1}{2}}(\frac{\frac{-1}{2}(8 + 0)}{(8x + 8)^{\frac{3}{2}}})x^{\frac{1}{2}} + \frac{8^{\frac{1}{2}}*\frac{1}{2}}{(8x + 8)^{\frac{1}{2}}x^{\frac{1}{2}}}\\=&\frac{8^{\frac{1}{2}}}{2(8x + 8)^{\frac{1}{2}}x^{\frac{1}{2}}} - \frac{4}{(8x + 1)^{\frac{3}{2}}} - \frac{4*8^{\frac{1}{2}}x^{\frac{1}{2}}}{(8x + 8)^{\frac{3}{2}}} - \frac{1}{2(x + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{8^{\frac{1}{2}}}{2(8x + 8)^{\frac{1}{2}}x^{\frac{1}{2}}} - \frac{4}{(8x + 1)^{\frac{3}{2}}} - \frac{4*8^{\frac{1}{2}}x^{\frac{1}{2}}}{(8x + 8)^{\frac{3}{2}}} - \frac{1}{2(x + 1)^{\frac{3}{2}}}\right)}{dx}\\=&\frac{8^{\frac{1}{2}}(\frac{\frac{-1}{2}(8 + 0)}{(8x + 8)^{\frac{3}{2}}})}{2x^{\frac{1}{2}}} + \frac{8^{\frac{1}{2}}*\frac{-1}{2}}{2(8x + 8)^{\frac{1}{2}}x^{\frac{3}{2}}} - 4(\frac{\frac{-3}{2}(8 + 0)}{(8x + 1)^{\frac{5}{2}}}) - 4*8^{\frac{1}{2}}(\frac{\frac{-3}{2}(8 + 0)}{(8x + 8)^{\frac{5}{2}}})x^{\frac{1}{2}} - \frac{4*8^{\frac{1}{2}}*\frac{1}{2}}{(8x + 8)^{\frac{3}{2}}x^{\frac{1}{2}}} - \frac{(\frac{\frac{-3}{2}(1 + 0)}{(x + 1)^{\frac{5}{2}}})}{2}\\=& - \frac{4*8^{\frac{1}{2}}}{(8x + 8)^{\frac{3}{2}}x^{\frac{1}{2}}} - \frac{8^{\frac{1}{2}}}{4(8x + 8)^{\frac{1}{2}}x^{\frac{3}{2}}} + \frac{48*8^{\frac{1}{2}}x^{\frac{1}{2}}}{(8x + 8)^{\frac{5}{2}}} + \frac{48}{(8x + 1)^{\frac{5}{2}}} + \frac{3}{4(x + 1)^{\frac{5}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( - \frac{4*8^{\frac{1}{2}}}{(8x + 8)^{\frac{3}{2}}x^{\frac{1}{2}}} - \frac{8^{\frac{1}{2}}}{4(8x + 8)^{\frac{1}{2}}x^{\frac{3}{2}}} + \frac{48*8^{\frac{1}{2}}x^{\frac{1}{2}}}{(8x + 8)^{\frac{5}{2}}} + \frac{48}{(8x + 1)^{\frac{5}{2}}} + \frac{3}{4(x + 1)^{\frac{5}{2}}}\right)}{dx}\\=& - \frac{4*8^{\frac{1}{2}}(\frac{\frac{-3}{2}(8 + 0)}{(8x + 8)^{\frac{5}{2}}})}{x^{\frac{1}{2}}} - \frac{4*8^{\frac{1}{2}}*\frac{-1}{2}}{(8x + 8)^{\frac{3}{2}}x^{\frac{3}{2}}} - \frac{8^{\frac{1}{2}}(\frac{\frac{-1}{2}(8 + 0)}{(8x + 8)^{\frac{3}{2}}})}{4x^{\frac{3}{2}}} - \frac{8^{\frac{1}{2}}*\frac{-3}{2}}{4(8x + 8)^{\frac{1}{2}}x^{\frac{5}{2}}} + 48*8^{\frac{1}{2}}(\frac{\frac{-5}{2}(8 + 0)}{(8x + 8)^{\frac{7}{2}}})x^{\frac{1}{2}} + \frac{48*8^{\frac{1}{2}}*\frac{1}{2}}{(8x + 8)^{\frac{5}{2}}x^{\frac{1}{2}}} + 48(\frac{\frac{-5}{2}(8 + 0)}{(8x + 1)^{\frac{7}{2}}}) + \frac{3(\frac{\frac{-5}{2}(1 + 0)}{(x + 1)^{\frac{7}{2}}})}{4}\\=&\frac{72*8^{\frac{1}{2}}}{(8x + 8)^{\frac{5}{2}}x^{\frac{1}{2}}} + \frac{3*8^{\frac{1}{2}}}{(8x + 8)^{\frac{3}{2}}x^{\frac{3}{2}}} + \frac{3*8^{\frac{1}{2}}}{8(8x + 8)^{\frac{1}{2}}x^{\frac{5}{2}}} - \frac{960*8^{\frac{1}{2}}x^{\frac{1}{2}}}{(8x + 8)^{\frac{7}{2}}} - \frac{960}{(8x + 1)^{\frac{7}{2}}} - \frac{15}{8(x + 1)^{\frac{7}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{72*8^{\frac{1}{2}}}{(8x + 8)^{\frac{5}{2}}x^{\frac{1}{2}}} + \frac{3*8^{\frac{1}{2}}}{(8x + 8)^{\frac{3}{2}}x^{\frac{3}{2}}} + \frac{3*8^{\frac{1}{2}}}{8(8x + 8)^{\frac{1}{2}}x^{\frac{5}{2}}} - \frac{960*8^{\frac{1}{2}}x^{\frac{1}{2}}}{(8x + 8)^{\frac{7}{2}}} - \frac{960}{(8x + 1)^{\frac{7}{2}}} - \frac{15}{8(x + 1)^{\frac{7}{2}}}\right)}{dx}\\=&\frac{72*8^{\frac{1}{2}}(\frac{\frac{-5}{2}(8 + 0)}{(8x + 8)^{\frac{7}{2}}})}{x^{\frac{1}{2}}} + \frac{72*8^{\frac{1}{2}}*\frac{-1}{2}}{(8x + 8)^{\frac{5}{2}}x^{\frac{3}{2}}} + \frac{3*8^{\frac{1}{2}}(\frac{\frac{-3}{2}(8 + 0)}{(8x + 8)^{\frac{5}{2}}})}{x^{\frac{3}{2}}} + \frac{3*8^{\frac{1}{2}}*\frac{-3}{2}}{(8x + 8)^{\frac{3}{2}}x^{\frac{5}{2}}} + \frac{3*8^{\frac{1}{2}}(\frac{\frac{-1}{2}(8 + 0)}{(8x + 8)^{\frac{3}{2}}})}{8x^{\frac{5}{2}}} + \frac{3*8^{\frac{1}{2}}*\frac{-5}{2}}{8(8x + 8)^{\frac{1}{2}}x^{\frac{7}{2}}} - 960*8^{\frac{1}{2}}(\frac{\frac{-7}{2}(8 + 0)}{(8x + 8)^{\frac{9}{2}}})x^{\frac{1}{2}} - \frac{960*8^{\frac{1}{2}}*\frac{1}{2}}{(8x + 8)^{\frac{7}{2}}x^{\frac{1}{2}}} - 960(\frac{\frac{-7}{2}(8 + 0)}{(8x + 1)^{\frac{9}{2}}}) - \frac{15(\frac{\frac{-7}{2}(1 + 0)}{(x + 1)^{\frac{9}{2}}})}{8}\\=& - \frac{1920*8^{\frac{1}{2}}}{(8x + 8)^{\frac{7}{2}}x^{\frac{1}{2}}} - \frac{72*8^{\frac{1}{2}}}{(8x + 8)^{\frac{5}{2}}x^{\frac{3}{2}}} - \frac{6*8^{\frac{1}{2}}}{(8x + 8)^{\frac{3}{2}}x^{\frac{5}{2}}} - \frac{15*8^{\frac{1}{2}}}{16(8x + 8)^{\frac{1}{2}}x^{\frac{7}{2}}} + \frac{26880*8^{\frac{1}{2}}x^{\frac{1}{2}}}{(8x + 8)^{\frac{9}{2}}} + \frac{26880}{(8x + 1)^{\frac{9}{2}}} + \frac{105}{16(x + 1)^{\frac{9}{2}}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {(1 + x)}^{(\frac{-1}{2})} + {(1 + 8x)}^{(\frac{-1}{2})} + {(\frac{8x}{(8x + 8)})}^{\frac{1}{2}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{1}{(x + 1)^{\frac{1}{2}}} + \frac{1}{(8x + 1)^{\frac{1}{2}}} + \frac{8^{\frac{1}{2}}x^{\frac{1}{2}}}{(8x + 8)^{\frac{1}{2}}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{1}{(x + 1)^{\frac{1}{2}}} + \frac{1}{(8x + 1)^{\frac{1}{2}}} + \frac{8^{\frac{1}{2}}x^{\frac{1}{2}}}{(8x + 8)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(1 + 0)}{(x + 1)^{\frac{3}{2}}}) + (\frac{\frac{-1}{2}(8 + 0)}{(8x + 1)^{\frac{3}{2}}}) + 8^{\frac{1}{2}}(\frac{\frac{-1}{2}(8 + 0)}{(8x + 8)^{\frac{3}{2}}})x^{\frac{1}{2}} + \frac{8^{\frac{1}{2}}*\frac{1}{2}}{(8x + 8)^{\frac{1}{2}}x^{\frac{1}{2}}}\\=&\frac{8^{\frac{1}{2}}}{2(8x + 8)^{\frac{1}{2}}x^{\frac{1}{2}}} - \frac{4}{(8x + 1)^{\frac{3}{2}}} - \frac{4*8^{\frac{1}{2}}x^{\frac{1}{2}}}{(8x + 8)^{\frac{3}{2}}} - \frac{1}{2(x + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{8^{\frac{1}{2}}}{2(8x + 8)^{\frac{1}{2}}x^{\frac{1}{2}}} - \frac{4}{(8x + 1)^{\frac{3}{2}}} - \frac{4*8^{\frac{1}{2}}x^{\frac{1}{2}}}{(8x + 8)^{\frac{3}{2}}} - \frac{1}{2(x + 1)^{\frac{3}{2}}}\right)}{dx}\\=&\frac{8^{\frac{1}{2}}(\frac{\frac{-1}{2}(8 + 0)}{(8x + 8)^{\frac{3}{2}}})}{2x^{\frac{1}{2}}} + \frac{8^{\frac{1}{2}}*\frac{-1}{2}}{2(8x + 8)^{\frac{1}{2}}x^{\frac{3}{2}}} - 4(\frac{\frac{-3}{2}(8 + 0)}{(8x + 1)^{\frac{5}{2}}}) - 4*8^{\frac{1}{2}}(\frac{\frac{-3}{2}(8 + 0)}{(8x + 8)^{\frac{5}{2}}})x^{\frac{1}{2}} - \frac{4*8^{\frac{1}{2}}*\frac{1}{2}}{(8x + 8)^{\frac{3}{2}}x^{\frac{1}{2}}} - \frac{(\frac{\frac{-3}{2}(1 + 0)}{(x + 1)^{\frac{5}{2}}})}{2}\\=& - \frac{4*8^{\frac{1}{2}}}{(8x + 8)^{\frac{3}{2}}x^{\frac{1}{2}}} - \frac{8^{\frac{1}{2}}}{4(8x + 8)^{\frac{1}{2}}x^{\frac{3}{2}}} + \frac{48*8^{\frac{1}{2}}x^{\frac{1}{2}}}{(8x + 8)^{\frac{5}{2}}} + \frac{48}{(8x + 1)^{\frac{5}{2}}} + \frac{3}{4(x + 1)^{\frac{5}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( - \frac{4*8^{\frac{1}{2}}}{(8x + 8)^{\frac{3}{2}}x^{\frac{1}{2}}} - \frac{8^{\frac{1}{2}}}{4(8x + 8)^{\frac{1}{2}}x^{\frac{3}{2}}} + \frac{48*8^{\frac{1}{2}}x^{\frac{1}{2}}}{(8x + 8)^{\frac{5}{2}}} + \frac{48}{(8x + 1)^{\frac{5}{2}}} + \frac{3}{4(x + 1)^{\frac{5}{2}}}\right)}{dx}\\=& - \frac{4*8^{\frac{1}{2}}(\frac{\frac{-3}{2}(8 + 0)}{(8x + 8)^{\frac{5}{2}}})}{x^{\frac{1}{2}}} - \frac{4*8^{\frac{1}{2}}*\frac{-1}{2}}{(8x + 8)^{\frac{3}{2}}x^{\frac{3}{2}}} - \frac{8^{\frac{1}{2}}(\frac{\frac{-1}{2}(8 + 0)}{(8x + 8)^{\frac{3}{2}}})}{4x^{\frac{3}{2}}} - \frac{8^{\frac{1}{2}}*\frac{-3}{2}}{4(8x + 8)^{\frac{1}{2}}x^{\frac{5}{2}}} + 48*8^{\frac{1}{2}}(\frac{\frac{-5}{2}(8 + 0)}{(8x + 8)^{\frac{7}{2}}})x^{\frac{1}{2}} + \frac{48*8^{\frac{1}{2}}*\frac{1}{2}}{(8x + 8)^{\frac{5}{2}}x^{\frac{1}{2}}} + 48(\frac{\frac{-5}{2}(8 + 0)}{(8x + 1)^{\frac{7}{2}}}) + \frac{3(\frac{\frac{-5}{2}(1 + 0)}{(x + 1)^{\frac{7}{2}}})}{4}\\=&\frac{72*8^{\frac{1}{2}}}{(8x + 8)^{\frac{5}{2}}x^{\frac{1}{2}}} + \frac{3*8^{\frac{1}{2}}}{(8x + 8)^{\frac{3}{2}}x^{\frac{3}{2}}} + \frac{3*8^{\frac{1}{2}}}{8(8x + 8)^{\frac{1}{2}}x^{\frac{5}{2}}} - \frac{960*8^{\frac{1}{2}}x^{\frac{1}{2}}}{(8x + 8)^{\frac{7}{2}}} - \frac{960}{(8x + 1)^{\frac{7}{2}}} - \frac{15}{8(x + 1)^{\frac{7}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{72*8^{\frac{1}{2}}}{(8x + 8)^{\frac{5}{2}}x^{\frac{1}{2}}} + \frac{3*8^{\frac{1}{2}}}{(8x + 8)^{\frac{3}{2}}x^{\frac{3}{2}}} + \frac{3*8^{\frac{1}{2}}}{8(8x + 8)^{\frac{1}{2}}x^{\frac{5}{2}}} - \frac{960*8^{\frac{1}{2}}x^{\frac{1}{2}}}{(8x + 8)^{\frac{7}{2}}} - \frac{960}{(8x + 1)^{\frac{7}{2}}} - \frac{15}{8(x + 1)^{\frac{7}{2}}}\right)}{dx}\\=&\frac{72*8^{\frac{1}{2}}(\frac{\frac{-5}{2}(8 + 0)}{(8x + 8)^{\frac{7}{2}}})}{x^{\frac{1}{2}}} + \frac{72*8^{\frac{1}{2}}*\frac{-1}{2}}{(8x + 8)^{\frac{5}{2}}x^{\frac{3}{2}}} + \frac{3*8^{\frac{1}{2}}(\frac{\frac{-3}{2}(8 + 0)}{(8x + 8)^{\frac{5}{2}}})}{x^{\frac{3}{2}}} + \frac{3*8^{\frac{1}{2}}*\frac{-3}{2}}{(8x + 8)^{\frac{3}{2}}x^{\frac{5}{2}}} + \frac{3*8^{\frac{1}{2}}(\frac{\frac{-1}{2}(8 + 0)}{(8x + 8)^{\frac{3}{2}}})}{8x^{\frac{5}{2}}} + \frac{3*8^{\frac{1}{2}}*\frac{-5}{2}}{8(8x + 8)^{\frac{1}{2}}x^{\frac{7}{2}}} - 960*8^{\frac{1}{2}}(\frac{\frac{-7}{2}(8 + 0)}{(8x + 8)^{\frac{9}{2}}})x^{\frac{1}{2}} - \frac{960*8^{\frac{1}{2}}*\frac{1}{2}}{(8x + 8)^{\frac{7}{2}}x^{\frac{1}{2}}} - 960(\frac{\frac{-7}{2}(8 + 0)}{(8x + 1)^{\frac{9}{2}}}) - \frac{15(\frac{\frac{-7}{2}(1 + 0)}{(x + 1)^{\frac{9}{2}}})}{8}\\=& - \frac{1920*8^{\frac{1}{2}}}{(8x + 8)^{\frac{7}{2}}x^{\frac{1}{2}}} - \frac{72*8^{\frac{1}{2}}}{(8x + 8)^{\frac{5}{2}}x^{\frac{3}{2}}} - \frac{6*8^{\frac{1}{2}}}{(8x + 8)^{\frac{3}{2}}x^{\frac{5}{2}}} - \frac{15*8^{\frac{1}{2}}}{16(8x + 8)^{\frac{1}{2}}x^{\frac{7}{2}}} + \frac{26880*8^{\frac{1}{2}}x^{\frac{1}{2}}}{(8x + 8)^{\frac{9}{2}}} + \frac{26880}{(8x + 1)^{\frac{9}{2}}} + \frac{105}{16(x + 1)^{\frac{9}{2}}}\\ \end{split}\end{equation} \]