There are 1 questions in this calculation: for each question, the 4 derivative of n is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {(1 + \frac{1}{n})}^{n}\ with\ respect\ to\ n：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = (\frac{1}{n} + 1)^{n}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( (\frac{1}{n} + 1)^{n}\right)}{dn}\\=&((\frac{1}{n} + 1)^{n}((1)ln(\frac{1}{n} + 1) + \frac{(n)(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)}))\\=&(\frac{1}{n} + 1)^{n}ln(\frac{1}{n} + 1) - \frac{(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)n}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( (\frac{1}{n} + 1)^{n}ln(\frac{1}{n} + 1) - \frac{(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)n}\right)}{dn}\\=&((\frac{1}{n} + 1)^{n}((1)ln(\frac{1}{n} + 1) + \frac{(n)(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)}))ln(\frac{1}{n} + 1) + \frac{(\frac{1}{n} + 1)^{n}(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)} - \frac{(\frac{-(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)^{2}})(\frac{1}{n} + 1)^{n}}{n} - \frac{-(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)n^{2}} - \frac{((\frac{1}{n} + 1)^{n}((1)ln(\frac{1}{n} + 1) + \frac{(n)(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)}))}{(\frac{1}{n} + 1)n}\\=&(\frac{1}{n} + 1)^{n}ln^{2}(\frac{1}{n} + 1) - \frac{2(\frac{1}{n} + 1)^{n}ln(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)n} - \frac{(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{2}n^{3}} + \frac{(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{2}n^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( (\frac{1}{n} + 1)^{n}ln^{2}(\frac{1}{n} + 1) - \frac{2(\frac{1}{n} + 1)^{n}ln(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)n} - \frac{(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{2}n^{3}} + \frac{(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{2}n^{2}}\right)}{dn}\\=&((\frac{1}{n} + 1)^{n}((1)ln(\frac{1}{n} + 1) + \frac{(n)(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)}))ln^{2}(\frac{1}{n} + 1) + \frac{(\frac{1}{n} + 1)^{n}*2ln(\frac{1}{n} + 1)(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)} - \frac{2(\frac{-(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)^{2}})(\frac{1}{n} + 1)^{n}ln(\frac{1}{n} + 1)}{n} - \frac{2*-(\frac{1}{n} + 1)^{n}ln(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)n^{2}} - \frac{2((\frac{1}{n} + 1)^{n}((1)ln(\frac{1}{n} + 1) + \frac{(n)(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)}))ln(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)n} - \frac{2(\frac{1}{n} + 1)^{n}(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)n(\frac{1}{n} + 1)} - \frac{(\frac{-2(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)^{3}})(\frac{1}{n} + 1)^{n}}{n^{3}} - \frac{-3(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{2}n^{4}} - \frac{((\frac{1}{n} + 1)^{n}((1)ln(\frac{1}{n} + 1) + \frac{(n)(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)}))}{(\frac{1}{n} + 1)^{2}n^{3}} + \frac{(\frac{-2(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)^{3}})(\frac{1}{n} + 1)^{n}}{n^{2}} + \frac{-2(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{2}n^{3}} + \frac{((\frac{1}{n} + 1)^{n}((1)ln(\frac{1}{n} + 1) + \frac{(n)(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)}))}{(\frac{1}{n} + 1)^{2}n^{2}}\\=&(\frac{1}{n} + 1)^{n}ln^{3}(\frac{1}{n} + 1) - \frac{3(\frac{1}{n} + 1)^{n}ln^{2}(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)n} - \frac{3(\frac{1}{n} + 1)^{n}ln(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)^{2}n^{3}} + \frac{3(\frac{1}{n} + 1)^{n}ln(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)^{2}n^{2}} - \frac{2(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{3}n^{5}} + \frac{3(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{2}n^{4}} + \frac{3(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{3}n^{4}} - \frac{(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{3}n^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( (\frac{1}{n} + 1)^{n}ln^{3}(\frac{1}{n} + 1) - \frac{3(\frac{1}{n} + 1)^{n}ln^{2}(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)n} - \frac{3(\frac{1}{n} + 1)^{n}ln(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)^{2}n^{3}} + \frac{3(\frac{1}{n} + 1)^{n}ln(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)^{2}n^{2}} - \frac{2(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{3}n^{5}} + \frac{3(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{2}n^{4}} + \frac{3(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{3}n^{4}} - \frac{(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{3}n^{3}}\right)}{dn}\\=&((\frac{1}{n} + 1)^{n}((1)ln(\frac{1}{n} + 1) + \frac{(n)(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)}))ln^{3}(\frac{1}{n} + 1) + \frac{(\frac{1}{n} + 1)^{n}*3ln^{2}(\frac{1}{n} + 1)(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)} - \frac{3(\frac{-(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)^{2}})(\frac{1}{n} + 1)^{n}ln^{2}(\frac{1}{n} + 1)}{n} - \frac{3*-(\frac{1}{n} + 1)^{n}ln^{2}(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)n^{2}} - \frac{3((\frac{1}{n} + 1)^{n}((1)ln(\frac{1}{n} + 1) + \frac{(n)(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)}))ln^{2}(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)n} - \frac{3(\frac{1}{n} + 1)^{n}*2ln(\frac{1}{n} + 1)(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)n(\frac{1}{n} + 1)} - \frac{3(\frac{-2(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)^{3}})(\frac{1}{n} + 1)^{n}ln(\frac{1}{n} + 1)}{n^{3}} - \frac{3*-3(\frac{1}{n} + 1)^{n}ln(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)^{2}n^{4}} - \frac{3((\frac{1}{n} + 1)^{n}((1)ln(\frac{1}{n} + 1) + \frac{(n)(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)}))ln(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)^{2}n^{3}} - \frac{3(\frac{1}{n} + 1)^{n}(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)^{2}n^{3}(\frac{1}{n} + 1)} + \frac{3(\frac{-2(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)^{3}})(\frac{1}{n} + 1)^{n}ln(\frac{1}{n} + 1)}{n^{2}} + \frac{3*-2(\frac{1}{n} + 1)^{n}ln(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)^{2}n^{3}} + \frac{3((\frac{1}{n} + 1)^{n}((1)ln(\frac{1}{n} + 1) + \frac{(n)(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)}))ln(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)^{2}n^{2}} + \frac{3(\frac{1}{n} + 1)^{n}(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)^{2}n^{2}(\frac{1}{n} + 1)} - \frac{2(\frac{-3(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)^{4}})(\frac{1}{n} + 1)^{n}}{n^{5}} - \frac{2*-5(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{3}n^{6}} - \frac{2((\frac{1}{n} + 1)^{n}((1)ln(\frac{1}{n} + 1) + \frac{(n)(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)}))}{(\frac{1}{n} + 1)^{3}n^{5}} + \frac{3(\frac{-2(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)^{3}})(\frac{1}{n} + 1)^{n}}{n^{4}} + \frac{3*-4(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{2}n^{5}} + \frac{3((\frac{1}{n} + 1)^{n}((1)ln(\frac{1}{n} + 1) + \frac{(n)(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)}))}{(\frac{1}{n} + 1)^{2}n^{4}} + \frac{3(\frac{-3(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)^{4}})(\frac{1}{n} + 1)^{n}}{n^{4}} + \frac{3*-4(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{3}n^{5}} + \frac{3((\frac{1}{n} + 1)^{n}((1)ln(\frac{1}{n} + 1) + \frac{(n)(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)}))}{(\frac{1}{n} + 1)^{3}n^{4}} - \frac{(\frac{-3(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)^{4}})(\frac{1}{n} + 1)^{n}}{n^{3}} - \frac{-3(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{3}n^{4}} - \frac{((\frac{1}{n} + 1)^{n}((1)ln(\frac{1}{n} + 1) + \frac{(n)(\frac{-1}{n^{2}} + 0)}{(\frac{1}{n} + 1)}))}{(\frac{1}{n} + 1)^{3}n^{3}}\\=&(\frac{1}{n} + 1)^{n}ln^{4}(\frac{1}{n} + 1) - \frac{4(\frac{1}{n} + 1)^{n}ln^{3}(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)n} - \frac{6(\frac{1}{n} + 1)^{n}ln^{2}(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)^{2}n^{3}} + \frac{6(\frac{1}{n} + 1)^{n}ln^{2}(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)^{2}n^{2}} - \frac{8(\frac{1}{n} + 1)^{n}ln(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)^{3}n^{5}} + \frac{12(\frac{1}{n} + 1)^{n}ln(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)^{2}n^{4}} + \frac{12(\frac{1}{n} + 1)^{n}ln(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)^{3}n^{4}} - \frac{4(\frac{1}{n} + 1)^{n}ln(\frac{1}{n} + 1)}{(\frac{1}{n} + 1)^{3}n^{3}} - \frac{12(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{3}n^{5}} - \frac{6(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{4}n^{7}} + \frac{16(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{3}n^{6}} + \frac{11(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{4}n^{6}} - \frac{12(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{2}n^{5}} - \frac{6(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{4}n^{5}} + \frac{(\frac{1}{n} + 1)^{n}}{(\frac{1}{n} + 1)^{4}n^{4}}\\ \end{split}\end{equation} \]

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