There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
Note that variables are case sensitive.\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ cos(x) + sin(x)({e}^{x} + {x}^{(\frac{e^{-1}}{arctan(x)} + arcsin(x) + 2)}{\frac{1}{x}}^{(3e^{x})})\ with\ respect\ to\ x:\\\end{split}\end{equation} \]
\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = cos(x) + {e}^{x}sin(x) + {x}^{(\frac{e^{-1}}{arctan(x)} + arcsin(x) + 2)}{\frac{1}{x}}^{(3e^{x})}sin(x)\\&\color{blue}{The\ first\ derivative\ function:}\\&\frac{d\left( cos(x) + {e}^{x}sin(x) + {x}^{(\frac{e^{-1}}{arctan(x)} + arcsin(x) + 2)}{\frac{1}{x}}^{(3e^{x})}sin(x)\right)}{dx}\\=&-sin(x) + ({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))sin(x) + {e}^{x}cos(x) + ({x}^{(\frac{e^{-1}}{arctan(x)} + arcsin(x) + 2)}((\frac{e^{-1}*0}{arctan(x)} + e^{-1}(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})}) + (\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}) + 0)ln(x) + \frac{(\frac{e^{-1}}{arctan(x)} + arcsin(x) + 2)(1)}{(x)})){\frac{1}{x}}^{(3e^{x})}sin(x) + {x}^{(\frac{e^{-1}}{arctan(x)} + arcsin(x) + 2)}({\frac{1}{x}}^{(3e^{x})}((3e^{x})ln(\frac{1}{x}) + \frac{(3e^{x})(\frac{-1}{x^{2}})}{(\frac{1}{x})}))sin(x) + {x}^{(\frac{e^{-1}}{arctan(x)} + arcsin(x) + 2)}{\frac{1}{x}}^{(3e^{x})}cos(x)\\=&-sin(x) + {e}^{x}sin(x) + {e}^{x}cos(x) - \frac{{x}^{(\frac{e^{-1}}{arctan(x)} + arcsin(x) + 2)}{\frac{1}{x}}^{(3e^{x})}e^{-1}ln(x)sin(x)}{(x^{2} + 1)arctan^{2}(x)} + \frac{{x}^{(\frac{e^{-1}}{arctan(x)} + arcsin(x) + 2)}{\frac{1}{x}}^{(3e^{x})}ln(x)sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{{x}^{(\frac{e^{-1}}{arctan(x)} + arcsin(x) + 2)}{\frac{1}{x}}^{(3e^{x})}e^{-1}sin(x)}{xarctan(x)} + \frac{{x}^{(\frac{e^{-1}}{arctan(x)} + arcsin(x) + 2)}{\frac{1}{x}}^{(3e^{x})}sin(x)arcsin(x)}{x} + \frac{2{x}^{(\frac{e^{-1}}{arctan(x)} + arcsin(x) + 2)}{\frac{1}{x}}^{(3e^{x})}sin(x)}{x} + 3{\frac{1}{x}}^{(3e^{x})}{x}^{(\frac{e^{-1}}{arctan(x)} + arcsin(x) + 2)}e^{x}ln(\frac{1}{x})sin(x) - \frac{3{\frac{1}{x}}^{(3e^{x})}{x}^{(\frac{e^{-1}}{arctan(x)} + arcsin(x) + 2)}e^{x}sin(x)}{x} + {\frac{1}{x}}^{(3e^{x})}{x}^{(\frac{e^{-1}}{arctan(x)} + arcsin(x) + 2)}cos(x)\\ \end{split}\end{equation} \]Your problem has not been solved here? Please go to the Hot Problems section!