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

\[ \begin{equation}\begin{split}[2/2]Find\ the\ first\ derivative\ of\ function\ \frac{a{e}^{(\frac{bx}{a})}(bsin(x) - arccos(x))}{({a}^{2} + {b}^{2})} + C\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{ab{e}^{(\frac{bx}{a})}sin(x)}{(a^{2} + b^{2})} - \frac{a{e}^{(\frac{bx}{a})}arccos(x)}{(a^{2} + b^{2})} + C\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{ab{e}^{(\frac{bx}{a})}sin(x)}{(a^{2} + b^{2})} - \frac{a{e}^{(\frac{bx}{a})}arccos(x)}{(a^{2} + b^{2})} + C\right)}{dx}\\=&(\frac{-(0 + 0)}{(a^{2} + b^{2})^{2}})ab{e}^{(\frac{bx}{a})}sin(x) + \frac{ab({e}^{(\frac{bx}{a})}((\frac{b}{a})ln(e) + \frac{(\frac{bx}{a})(0)}{(e)}))sin(x)}{(a^{2} + b^{2})} + \frac{ab{e}^{(\frac{bx}{a})}cos(x)}{(a^{2} + b^{2})} - (\frac{-(0 + 0)}{(a^{2} + b^{2})^{2}})a{e}^{(\frac{bx}{a})}arccos(x) - \frac{a({e}^{(\frac{bx}{a})}((\frac{b}{a})ln(e) + \frac{(\frac{bx}{a})(0)}{(e)}))arccos(x)}{(a^{2} + b^{2})} - \frac{a{e}^{(\frac{bx}{a})}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(a^{2} + b^{2})} + 0\\=&\frac{b^{2}{e}^{(\frac{bx}{a})}sin(x)}{(a^{2} + b^{2})} + \frac{ab{e}^{(\frac{bx}{a})}cos(x)}{(a^{2} + b^{2})} - \frac{b{e}^{(\frac{bx}{a})}arccos(x)}{(a^{2} + b^{2})} + \frac{a{e}^{(\frac{bx}{a})}}{(a^{2} + b^{2})(-x^{2} + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]

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