There are 1 questions in this calculation: for each question, the 2 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ second\ derivative\ of\ function\ {cot(x)}^{e^{x}}ln(x) + 4arctan(x) + arctan(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = {cot(x)}^{e^{x}}ln(x) + 5arctan(x)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {cot(x)}^{e^{x}}ln(x) + 5arctan(x)\right)}{dx}\\=&({cot(x)}^{e^{x}}((e^{x})ln(cot(x)) + \frac{(e^{x})(-csc^{2}(x))}{(cot(x))}))ln(x) + \frac{{cot(x)}^{e^{x}}}{(x)} + 5(\frac{(1)}{(1 + (x)^{2})})\\=&{cot(x)}^{e^{x}}e^{x}ln(cot(x))ln(x) - \frac{{cot(x)}^{e^{x}}e^{x}ln(x)csc^{2}(x)}{cot(x)} + \frac{{cot(x)}^{e^{x}}}{x} + \frac{5}{(x^{2} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( {cot(x)}^{e^{x}}e^{x}ln(cot(x))ln(x) - \frac{{cot(x)}^{e^{x}}e^{x}ln(x)csc^{2}(x)}{cot(x)} + \frac{{cot(x)}^{e^{x}}}{x} + \frac{5}{(x^{2} + 1)}\right)}{dx}\\=&({cot(x)}^{e^{x}}((e^{x})ln(cot(x)) + \frac{(e^{x})(-csc^{2}(x))}{(cot(x))}))e^{x}ln(cot(x))ln(x) + {cot(x)}^{e^{x}}e^{x}ln(cot(x))ln(x) + \frac{{cot(x)}^{e^{x}}e^{x}*-csc^{2}(x)ln(x)}{(cot(x))} + \frac{{cot(x)}^{e^{x}}e^{x}ln(cot(x))}{(x)} - \frac{({cot(x)}^{e^{x}}((e^{x})ln(cot(x)) + \frac{(e^{x})(-csc^{2}(x))}{(cot(x))}))e^{x}ln(x)csc^{2}(x)}{cot(x)} - \frac{{cot(x)}^{e^{x}}e^{x}ln(x)csc^{2}(x)}{cot(x)} - \frac{{cot(x)}^{e^{x}}e^{x}csc^{2}(x)}{(x)cot(x)} - \frac{{cot(x)}^{e^{x}}e^{x}ln(x)csc^{2}(x)csc^{2}(x)}{cot^{2}(x)} - \frac{{cot(x)}^{e^{x}}e^{x}ln(x)*-2csc^{2}(x)cot(x)}{cot(x)} + \frac{-{cot(x)}^{e^{x}}}{x^{2}} + \frac{({cot(x)}^{e^{x}}((e^{x})ln(cot(x)) + \frac{(e^{x})(-csc^{2}(x))}{(cot(x))}))}{x} + 5(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})\\=&\frac{-2{cot(x)}^{e^{x}}e^{{x}*{2}}ln(cot(x))ln(x)csc^{2}(x)}{cot(x)} + {cot(x)}^{e^{x}}e^{{x}*{2}}ln^{2}(cot(x))ln(x) + {cot(x)}^{e^{x}}e^{x}ln(cot(x))ln(x) - \frac{2{cot(x)}^{e^{x}}e^{x}ln(x)csc^{2}(x)}{cot(x)} + \frac{2{cot(x)}^{e^{x}}e^{x}ln(cot(x))}{x} + \frac{{cot(x)}^{e^{x}}e^{{x}*{2}}ln(x)csc^{4}(x)}{cot^{2}(x)} - \frac{2{cot(x)}^{e^{x}}e^{x}csc^{2}(x)}{xcot(x)} - \frac{{cot(x)}^{e^{x}}e^{x}ln(x)csc^{4}(x)}{cot^{2}(x)} + 2{cot(x)}^{e^{x}}e^{x}ln(x)csc^{2}(x) - \frac{{cot(x)}^{e^{x}}}{x^{2}} - \frac{10x}{(x^{2} + 1)^{2}}\\ \end{split}\end{equation} \]

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