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

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