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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {o}^{(\frac{2}{3} + \frac{9}{10}sqrt(8 - {x}^{(2sin(ae^{x}))}))}\ with\ respect\ to\ o：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = {o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}\right)}{do}\\=&({o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}((\frac{\frac{9}{10}(-({x}^{(2sin(ae^{x}))}((2cos(ae^{x})ae^{x}*0)ln(x) + \frac{(2sin(ae^{x}))(0)}{(x)})) + 0)*\frac{1}{2}}{(-{x}^{(2sin(ae^{x}))} + 8)^{\frac{1}{2}}} + 0)ln(o) + \frac{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})(1)}{(o)}))\\=&\frac{9{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)}{10o} + \frac{2{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}}{3o}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{9{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)}{10o} + \frac{2{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}}{3o}\right)}{do}\\=&\frac{9*-{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)}{10o^{2}} + \frac{9({o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}((\frac{\frac{9}{10}(-({x}^{(2sin(ae^{x}))}((2cos(ae^{x})ae^{x}*0)ln(x) + \frac{(2sin(ae^{x}))(0)}{(x)})) + 0)*\frac{1}{2}}{(-{x}^{(2sin(ae^{x}))} + 8)^{\frac{1}{2}}} + 0)ln(o) + \frac{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})(1)}{(o)}))sqrt(-{x}^{(2sin(ae^{x}))} + 8)}{10o} + \frac{9{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}(-({x}^{(2sin(ae^{x}))}((2cos(ae^{x})ae^{x}*0)ln(x) + \frac{(2sin(ae^{x}))(0)}{(x)})) + 0)*\frac{1}{2}}{10o(-{x}^{(2sin(ae^{x}))} + 8)^{\frac{1}{2}}} + \frac{2*-{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}}{3o^{2}} + \frac{2({o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}((\frac{\frac{9}{10}(-({x}^{(2sin(ae^{x}))}((2cos(ae^{x})ae^{x}*0)ln(x) + \frac{(2sin(ae^{x}))(0)}{(x)})) + 0)*\frac{1}{2}}{(-{x}^{(2sin(ae^{x}))} + 8)^{\frac{1}{2}}} + 0)ln(o) + \frac{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})(1)}{(o)}))}{3o}\\=&\frac{3{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)}{10o^{2}} + \frac{81{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)^{2}}{100o^{2}} - \frac{2{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}}{9o^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{3{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)}{10o^{2}} + \frac{81{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)^{2}}{100o^{2}} - \frac{2{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}}{9o^{2}}\right)}{do}\\=&\frac{3*-2{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)}{10o^{3}} + \frac{3({o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}((\frac{\frac{9}{10}(-({x}^{(2sin(ae^{x}))}((2cos(ae^{x})ae^{x}*0)ln(x) + \frac{(2sin(ae^{x}))(0)}{(x)})) + 0)*\frac{1}{2}}{(-{x}^{(2sin(ae^{x}))} + 8)^{\frac{1}{2}}} + 0)ln(o) + \frac{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})(1)}{(o)}))sqrt(-{x}^{(2sin(ae^{x}))} + 8)}{10o^{2}} + \frac{3{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}(-({x}^{(2sin(ae^{x}))}((2cos(ae^{x})ae^{x}*0)ln(x) + \frac{(2sin(ae^{x}))(0)}{(x)})) + 0)*\frac{1}{2}}{10o^{2}(-{x}^{(2sin(ae^{x}))} + 8)^{\frac{1}{2}}} + \frac{81*-2{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)^{2}}{100o^{3}} + \frac{81({o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}((\frac{\frac{9}{10}(-({x}^{(2sin(ae^{x}))}((2cos(ae^{x})ae^{x}*0)ln(x) + \frac{(2sin(ae^{x}))(0)}{(x)})) + 0)*\frac{1}{2}}{(-{x}^{(2sin(ae^{x}))} + 8)^{\frac{1}{2}}} + 0)ln(o) + \frac{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})(1)}{(o)}))sqrt(-{x}^{(2sin(ae^{x}))} + 8)^{2}}{100o^{2}} + \frac{81{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}*2(-{x}^{(2sin(ae^{x}))} + 8)^{\frac{1}{2}}(-({x}^{(2sin(ae^{x}))}((2cos(ae^{x})ae^{x}*0)ln(x) + \frac{(2sin(ae^{x}))(0)}{(x)})) + 0)*\frac{1}{2}}{100o^{2}(-{x}^{(2sin(ae^{x}))} + 8)^{\frac{1}{2}}} - \frac{2*-2{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}}{9o^{3}} - \frac{2({o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}((\frac{\frac{9}{10}(-({x}^{(2sin(ae^{x}))}((2cos(ae^{x})ae^{x}*0)ln(x) + \frac{(2sin(ae^{x}))(0)}{(x)})) + 0)*\frac{1}{2}}{(-{x}^{(2sin(ae^{x}))} + 8)^{\frac{1}{2}}} + 0)ln(o) + \frac{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})(1)}{(o)}))}{9o^{2}}\\=&\frac{-3{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)}{5o^{3}} - \frac{81{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)^{2}}{100o^{3}} + \frac{729{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)^{3}}{1000o^{3}} + \frac{8{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}}{27o^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-3{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)}{5o^{3}} - \frac{81{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)^{2}}{100o^{3}} + \frac{729{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)^{3}}{1000o^{3}} + \frac{8{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}}{27o^{3}}\right)}{do}\\=&\frac{-3*-3{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)}{5o^{4}} - \frac{3({o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}((\frac{\frac{9}{10}(-({x}^{(2sin(ae^{x}))}((2cos(ae^{x})ae^{x}*0)ln(x) + \frac{(2sin(ae^{x}))(0)}{(x)})) + 0)*\frac{1}{2}}{(-{x}^{(2sin(ae^{x}))} + 8)^{\frac{1}{2}}} + 0)ln(o) + \frac{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})(1)}{(o)}))sqrt(-{x}^{(2sin(ae^{x}))} + 8)}{5o^{3}} - \frac{3{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}(-({x}^{(2sin(ae^{x}))}((2cos(ae^{x})ae^{x}*0)ln(x) + \frac{(2sin(ae^{x}))(0)}{(x)})) + 0)*\frac{1}{2}}{5o^{3}(-{x}^{(2sin(ae^{x}))} + 8)^{\frac{1}{2}}} - \frac{81*-3{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)^{2}}{100o^{4}} - \frac{81({o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}((\frac{\frac{9}{10}(-({x}^{(2sin(ae^{x}))}((2cos(ae^{x})ae^{x}*0)ln(x) + \frac{(2sin(ae^{x}))(0)}{(x)})) + 0)*\frac{1}{2}}{(-{x}^{(2sin(ae^{x}))} + 8)^{\frac{1}{2}}} + 0)ln(o) + \frac{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})(1)}{(o)}))sqrt(-{x}^{(2sin(ae^{x}))} + 8)^{2}}{100o^{3}} - \frac{81{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}*2(-{x}^{(2sin(ae^{x}))} + 8)^{\frac{1}{2}}(-({x}^{(2sin(ae^{x}))}((2cos(ae^{x})ae^{x}*0)ln(x) + \frac{(2sin(ae^{x}))(0)}{(x)})) + 0)*\frac{1}{2}}{100o^{3}(-{x}^{(2sin(ae^{x}))} + 8)^{\frac{1}{2}}} + \frac{729*-3{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)^{3}}{1000o^{4}} + \frac{729({o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}((\frac{\frac{9}{10}(-({x}^{(2sin(ae^{x}))}((2cos(ae^{x})ae^{x}*0)ln(x) + \frac{(2sin(ae^{x}))(0)}{(x)})) + 0)*\frac{1}{2}}{(-{x}^{(2sin(ae^{x}))} + 8)^{\frac{1}{2}}} + 0)ln(o) + \frac{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})(1)}{(o)}))sqrt(-{x}^{(2sin(ae^{x}))} + 8)^{3}}{1000o^{3}} + \frac{729{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}*3(-{x}^{(2sin(ae^{x}))} + 8)(-({x}^{(2sin(ae^{x}))}((2cos(ae^{x})ae^{x}*0)ln(x) + \frac{(2sin(ae^{x}))(0)}{(x)})) + 0)*\frac{1}{2}}{1000o^{3}(-{x}^{(2sin(ae^{x}))} + 8)^{\frac{1}{2}}} + \frac{8*-3{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}}{27o^{4}} + \frac{8({o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}((\frac{\frac{9}{10}(-({x}^{(2sin(ae^{x}))}((2cos(ae^{x})ae^{x}*0)ln(x) + \frac{(2sin(ae^{x}))(0)}{(x)})) + 0)*\frac{1}{2}}{(-{x}^{(2sin(ae^{x}))} + 8)^{\frac{1}{2}}} + 0)ln(o) + \frac{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})(1)}{(o)}))}{27o^{3}}\\=&\frac{5{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)}{3o^{4}} + \frac{27{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)^{2}}{20o^{4}} - \frac{243{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)^{3}}{100o^{4}} + \frac{6561{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}sqrt(-{x}^{(2sin(ae^{x}))} + 8)^{4}}{10000o^{4}} - \frac{56{o}^{(\frac{9}{10}sqrt(-{x}^{(2sin(ae^{x}))} + 8) + \frac{2}{3})}}{81o^{4}}\\ \end{split}\end{equation} \]