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 o is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ 5{o}^{{(2 - 4({o}^{4} - 1))}^{\frac{1}{2}}}\ with\ respect\ to\ o：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = 5{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( 5{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}\right)}{do}\\=&5({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))\\=&\frac{-40o^{3}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{5(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-40o^{3}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{5(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o}\right)}{do}\\=&-40(\frac{\frac{-1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{3}{2}}})o^{3}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - \frac{40*3o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{40o^{3}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{40o^{3}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}(o)} + \frac{5(\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}){o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o} + \frac{5(-4o^{4} + 6)^{\frac{1}{2}}*-{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{2}} + \frac{5(-4o^{4} + 6)^{\frac{1}{2}}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))}{o}\\=&\frac{-320o^{6}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{120o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{320o^{6}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)} - 80o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - \frac{80o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{5(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{2}} - 20o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{30{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-320o^{6}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{120o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{320o^{6}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)} - 80o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - \frac{80o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{5(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{2}} - 20o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{30{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{2}}\right)}{do}\\=&-320(\frac{\frac{-3}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{5}{2}}})o^{6}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - \frac{320*6o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{320o^{6}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{320o^{6}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{3}{2}}(o)} - 120(\frac{\frac{-1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{3}{2}}})o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - \frac{120*2o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{120o^{2}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{120o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}(o)} + 320(\frac{-(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{2}})o^{6}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o) + \frac{320*6o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)} + \frac{320o^{6}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln^{2}(o)}{(-4o^{4} + 6)} + \frac{320o^{6}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}*2ln(o)}{(-4o^{4} + 6)(o)} - 80*2o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - 80o^{2}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o) - \frac{80o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(o)} - 80(\frac{\frac{-1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{3}{2}}})o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{80*2o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{80o^{2}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{5(\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}){o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{2}} - \frac{5(-4o^{4} + 6)^{\frac{1}{2}}*-2{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{3}} - \frac{5(-4o^{4} + 6)^{\frac{1}{2}}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))}{o^{2}} - 20*2o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} - 20o^{2}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)})) + \frac{30*-2{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{3}} + \frac{30({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))}{o^{2}}\\=&\frac{-7680o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{5}{2}}} - \frac{2880o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} + \frac{7680o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)^{2}} + \frac{2880o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)} - \frac{480o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{2560o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{3}(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - 240o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) + \frac{960o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)} + \frac{960o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - 80(-4o^{4} + 6)^{\frac{1}{2}}o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) + \frac{160o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{960o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{3}{2}}} - 180o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{40(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{3}} - \frac{90{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{3}} - \frac{240o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}} - 20(-4o^{4} + 6)^{\frac{1}{2}}o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-7680o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{5}{2}}} - \frac{2880o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} + \frac{7680o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)^{2}} + \frac{2880o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)} - \frac{480o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{2560o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{3}(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - 240o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) + \frac{960o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)} + \frac{960o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - 80(-4o^{4} + 6)^{\frac{1}{2}}o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) + \frac{160o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{960o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{3}{2}}} - 180o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{40(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{3}} - \frac{90{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{3}} - \frac{240o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}} - 20(-4o^{4} + 6)^{\frac{1}{2}}o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}\right)}{do}\\=&-7680(\frac{\frac{-5}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{7}{2}}})o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - \frac{7680*9o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{5}{2}}} - \frac{7680o^{9}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o)}{(-4o^{4} + 6)^{\frac{5}{2}}} - \frac{7680o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{5}{2}}(o)} - 2880(\frac{\frac{-3}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{5}{2}}})o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - \frac{2880*5o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{2880o^{5}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{2880o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{3}{2}}(o)} + 7680(\frac{-2(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{3}})o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o) + \frac{7680*9o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)^{2}} + \frac{7680o^{9}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln^{2}(o)}{(-4o^{4} + 6)^{2}} + \frac{7680o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}*2ln(o)}{(-4o^{4} + 6)^{2}(o)} + 2880(\frac{-(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{2}})o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o) + \frac{2880*5o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)} + \frac{2880o^{5}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln^{2}(o)}{(-4o^{4} + 6)} + \frac{2880o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}*2ln(o)}{(-4o^{4} + 6)(o)} - 480(\frac{\frac{-1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{3}{2}}})o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - \frac{480{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{480o({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{480o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}(o)} - 2560(\frac{\frac{-3}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{5}{2}}})o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{3}(o) - \frac{2560*9o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{3}(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{2560o^{9}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln^{3}(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{2560o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}*3ln^{2}(o)}{(-4o^{4} + 6)^{\frac{3}{2}}(o)} - 240{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - 240o({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o) - \frac{240o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(o)} + 960(\frac{-(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{2}})o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) + \frac{960*5o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)} + \frac{960o^{5}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o)}{(-4o^{4} + 6)} + \frac{960o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)(o)} + 960(\frac{\frac{-1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{3}{2}}})o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o) + \frac{960*5o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{960o^{5}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln^{2}(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{960o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}*2ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}(o)} - 80(\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}})o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - 80(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - 80(-4o^{4} + 6)^{\frac{1}{2}}o({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o) - \frac{80(-4o^{4} + 6)^{\frac{1}{2}}o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(o)} + 160(\frac{\frac{-1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{3}{2}}})o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) + \frac{160*5o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{160o^{5}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{160o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}(o)} - 960(\frac{\frac{-3}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{5}{2}}})o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{960*5o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{960o^{5}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))}{(-4o^{4} + 6)^{\frac{3}{2}}} - 180{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} - 180o({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)})) + \frac{40(\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}){o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{3}} + \frac{40(-4o^{4} + 6)^{\frac{1}{2}}*-3{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{4}} + \frac{40(-4o^{4} + 6)^{\frac{1}{2}}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))}{o^{3}} - \frac{90*-3{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{4}} - \frac{90({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))}{o^{3}} - 240(\frac{\frac{-1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{3}{2}}})o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{240{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{240o({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))}{(-4o^{4} + 6)^{\frac{1}{2}}} - 20(\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}})o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} - 20(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} - 20(-4o^{4} + 6)^{\frac{1}{2}}o({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))\\=&\frac{-307200o^{12}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{7}{2}}} - \frac{138240o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{5}{2}}} + \frac{307200o^{12}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)^{3}} + \frac{138240o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)^{2}} - \frac{18240o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} + \frac{18240o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)} - \frac{46080o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{3}(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} + \frac{9600o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)} - \frac{122880o^{12}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{3}(o)}{(-4o^{4} + 6)^{\frac{5}{2}}} + \frac{30720o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{2}} + \frac{20480o^{12}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{4}(o)}{(-4o^{4} + 6)^{2}} + \frac{240{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{10240o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{3}(o)}{(-4o^{4} + 6)} + 1600o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o) - 320(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) + \frac{9600o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{5760o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{1280o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)} - 1520{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - 940{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{1280o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{30720o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{5}{2}}} - \frac{9600o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{1040{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}} + 640o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - 280(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{320o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{210(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{4}} + \frac{510{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{4}} + 80o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ 5{o}^{{(2 - 4({o}^{4} - 1))}^{\frac{1}{2}}}\ with\ respect\ to\ o：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = 5{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( 5{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}\right)}{do}\\=&5({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))\\=&\frac{-40o^{3}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{5(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-40o^{3}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{5(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o}\right)}{do}\\=&-40(\frac{\frac{-1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{3}{2}}})o^{3}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - \frac{40*3o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{40o^{3}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{40o^{3}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}(o)} + \frac{5(\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}){o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o} + \frac{5(-4o^{4} + 6)^{\frac{1}{2}}*-{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{2}} + \frac{5(-4o^{4} + 6)^{\frac{1}{2}}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))}{o}\\=&\frac{-320o^{6}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{120o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{320o^{6}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)} - 80o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - \frac{80o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{5(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{2}} - 20o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{30{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-320o^{6}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{120o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{320o^{6}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)} - 80o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - \frac{80o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{5(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{2}} - 20o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{30{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{2}}\right)}{do}\\=&-320(\frac{\frac{-3}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{5}{2}}})o^{6}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - \frac{320*6o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{320o^{6}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{320o^{6}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{3}{2}}(o)} - 120(\frac{\frac{-1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{3}{2}}})o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - \frac{120*2o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{120o^{2}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{120o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}(o)} + 320(\frac{-(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{2}})o^{6}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o) + \frac{320*6o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)} + \frac{320o^{6}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln^{2}(o)}{(-4o^{4} + 6)} + \frac{320o^{6}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}*2ln(o)}{(-4o^{4} + 6)(o)} - 80*2o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - 80o^{2}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o) - \frac{80o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(o)} - 80(\frac{\frac{-1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{3}{2}}})o^{2}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{80*2o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{80o^{2}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{5(\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}){o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{2}} - \frac{5(-4o^{4} + 6)^{\frac{1}{2}}*-2{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{3}} - \frac{5(-4o^{4} + 6)^{\frac{1}{2}}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))}{o^{2}} - 20*2o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} - 20o^{2}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)})) + \frac{30*-2{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{3}} + \frac{30({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))}{o^{2}}\\=&\frac{-7680o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{5}{2}}} - \frac{2880o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} + \frac{7680o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)^{2}} + \frac{2880o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)} - \frac{480o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{2560o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{3}(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - 240o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) + \frac{960o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)} + \frac{960o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - 80(-4o^{4} + 6)^{\frac{1}{2}}o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) + \frac{160o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{960o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{3}{2}}} - 180o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{40(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{3}} - \frac{90{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{3}} - \frac{240o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}} - 20(-4o^{4} + 6)^{\frac{1}{2}}o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-7680o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{5}{2}}} - \frac{2880o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} + \frac{7680o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)^{2}} + \frac{2880o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)} - \frac{480o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{2560o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{3}(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - 240o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) + \frac{960o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)} + \frac{960o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - 80(-4o^{4} + 6)^{\frac{1}{2}}o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) + \frac{160o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{960o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{3}{2}}} - 180o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{40(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{3}} - \frac{90{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{3}} - \frac{240o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}} - 20(-4o^{4} + 6)^{\frac{1}{2}}o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}\right)}{do}\\=&-7680(\frac{\frac{-5}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{7}{2}}})o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - \frac{7680*9o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{5}{2}}} - \frac{7680o^{9}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o)}{(-4o^{4} + 6)^{\frac{5}{2}}} - \frac{7680o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{5}{2}}(o)} - 2880(\frac{\frac{-3}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{5}{2}}})o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - \frac{2880*5o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{2880o^{5}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{2880o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{3}{2}}(o)} + 7680(\frac{-2(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{3}})o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o) + \frac{7680*9o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)^{2}} + \frac{7680o^{9}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln^{2}(o)}{(-4o^{4} + 6)^{2}} + \frac{7680o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}*2ln(o)}{(-4o^{4} + 6)^{2}(o)} + 2880(\frac{-(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{2}})o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o) + \frac{2880*5o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)} + \frac{2880o^{5}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln^{2}(o)}{(-4o^{4} + 6)} + \frac{2880o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}*2ln(o)}{(-4o^{4} + 6)(o)} - 480(\frac{\frac{-1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{3}{2}}})o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - \frac{480{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{480o({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{480o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}(o)} - 2560(\frac{\frac{-3}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{5}{2}}})o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{3}(o) - \frac{2560*9o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{3}(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{2560o^{9}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln^{3}(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{2560o^{9}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}*3ln^{2}(o)}{(-4o^{4} + 6)^{\frac{3}{2}}(o)} - 240{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - 240o({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o) - \frac{240o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(o)} + 960(\frac{-(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{2}})o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) + \frac{960*5o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)} + \frac{960o^{5}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o)}{(-4o^{4} + 6)} + \frac{960o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)(o)} + 960(\frac{\frac{-1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{3}{2}}})o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o) + \frac{960*5o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{960o^{5}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln^{2}(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{960o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}*2ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}(o)} - 80(\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}})o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - 80(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - 80(-4o^{4} + 6)^{\frac{1}{2}}o({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o) - \frac{80(-4o^{4} + 6)^{\frac{1}{2}}o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(o)} + 160(\frac{\frac{-1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{3}{2}}})o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) + \frac{160*5o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{160o^{5}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{160o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}(o)} - 960(\frac{\frac{-3}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{5}{2}}})o^{5}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{960*5o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{960o^{5}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))}{(-4o^{4} + 6)^{\frac{3}{2}}} - 180{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} - 180o({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)})) + \frac{40(\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}){o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{3}} + \frac{40(-4o^{4} + 6)^{\frac{1}{2}}*-3{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{4}} + \frac{40(-4o^{4} + 6)^{\frac{1}{2}}({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))}{o^{3}} - \frac{90*-3{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{4}} - \frac{90({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))}{o^{3}} - 240(\frac{\frac{-1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{3}{2}}})o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{240{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{240o({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))}{(-4o^{4} + 6)^{\frac{1}{2}}} - 20(\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}})o{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} - 20(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} - 20(-4o^{4} + 6)^{\frac{1}{2}}o({o}^{(-4o^{4} + 6)^{\frac{1}{2}}}(((\frac{\frac{1}{2}(-4*4o^{3} + 0)}{(-4o^{4} + 6)^{\frac{1}{2}}}))ln(o) + \frac{((-4o^{4} + 6)^{\frac{1}{2}})(1)}{(o)}))\\=&\frac{-307200o^{12}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{7}{2}}} - \frac{138240o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{5}{2}}} + \frac{307200o^{12}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)^{3}} + \frac{138240o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)^{2}} - \frac{18240o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} + \frac{18240o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)} - \frac{46080o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{3}(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} + \frac{9600o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)} - \frac{122880o^{12}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{3}(o)}{(-4o^{4} + 6)^{\frac{5}{2}}} + \frac{30720o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{2}} + \frac{20480o^{12}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{4}(o)}{(-4o^{4} + 6)^{2}} + \frac{240{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{10240o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{3}(o)}{(-4o^{4} + 6)} + 1600o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o) - 320(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) + \frac{9600o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{5760o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{1280o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln^{2}(o)}{(-4o^{4} + 6)} - 1520{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - 940{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{1280o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o)}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{30720o^{8}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{5}{2}}} - \frac{9600o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{3}{2}}} - \frac{1040{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}} + 640o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}ln(o) - 280(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}} + \frac{320o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{(-4o^{4} + 6)^{\frac{1}{2}}} - \frac{210(-4o^{4} + 6)^{\frac{1}{2}}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{4}} + \frac{510{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}}{o^{4}} + 80o^{4}{o}^{(-4o^{4} + 6)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]