There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ (\frac{sqrt(sin(x))}{x}) + \frac{sin(x)}{x} + sin((\frac{sqrt(sin(x))}{x})) + sin(\frac{sin(x)}{x}) + sin({x}^{x}) + sin({x}^{20}) + sin({x}^{sin(x)})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{sqrt(sin(x))}{x} + \frac{sin(x)}{x} + sin(\frac{sqrt(sin(x))}{x}) + sin(\frac{sin(x)}{x}) + sin({x}^{x}) + sin(x^{20}) + sin({x}^{sin(x)})\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{sqrt(sin(x))}{x} + \frac{sin(x)}{x} + sin(\frac{sqrt(sin(x))}{x}) + sin(\frac{sin(x)}{x}) + sin({x}^{x}) + sin(x^{20}) + sin({x}^{sin(x)})\right)}{dx}\\=&\frac{-sqrt(sin(x))}{x^{2}} + \frac{cos(x)*\frac{1}{2}}{x(sin(x))^{\frac{1}{2}}} + \frac{-sin(x)}{x^{2}} + \frac{cos(x)}{x} + cos(\frac{sqrt(sin(x))}{x})(\frac{-sqrt(sin(x))}{x^{2}} + \frac{cos(x)*\frac{1}{2}}{x(sin(x))^{\frac{1}{2}}}) + cos(\frac{sin(x)}{x})(\frac{-sin(x)}{x^{2}} + \frac{cos(x)}{x}) + cos({x}^{x})({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + cos(x^{20})*20x^{19} + cos({x}^{sin(x)})({x}^{sin(x)}((cos(x))ln(x) + \frac{(sin(x))(1)}{(x)}))\\=&\frac{-sqrt(sin(x))}{x^{2}} + \frac{cos(\frac{sqrt(sin(x))}{x})cos(x)}{2xsin^{\frac{1}{2}}(x)} - \frac{sin(x)cos(\frac{sin(x)}{x})}{x^{2}} - \frac{cos(\frac{sqrt(sin(x))}{x})sqrt(sin(x))}{x^{2}} + \frac{cos(x)cos(\frac{sin(x)}{x})}{x} + \frac{cos(x)}{2xsin^{\frac{1}{2}}(x)} - \frac{sin(x)}{x^{2}} + \frac{cos(x)}{x} + {x}^{sin(x)}ln(x)cos(x)cos({x}^{sin(x)}) + {x}^{x}cos({x}^{x}) + 20x^{19}cos(x^{20}) + {x}^{x}ln(x)cos({x}^{x}) + \frac{{x}^{sin(x)}sin(x)cos({x}^{sin(x)})}{x}\\ \end{split}\end{equation} \]

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