There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
Note that variables are case sensitive.\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ \frac{({(\frac{(sqrt(5) + 1)}{2})}^{x} - cos(Pix){\frac{1}{(\frac{(sqrt(5) + 1)}{2})}}^{x})}{({(\frac{(sqrt(5) + 1)}{2})}^{x} - cos(Pix){\frac{1}{(\frac{(sqrt(5) + 1)}{2})}}^{(x + 2)})}\ with\ respect\ to\ x:\\\end{split}\end{equation} \]
\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = - \frac{{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{x}cos(Pix)}{(-{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}cos(Pix) + (\frac{1}{2}sqrt(5) + \frac{1}{2})^{x})} + \frac{(\frac{1}{2}sqrt(5) + \frac{1}{2})^{x}}{(-{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}cos(Pix) + (\frac{1}{2}sqrt(5) + \frac{1}{2})^{x})}\\&\color{blue}{The\ first\ derivative\ function:}\\&\frac{d\left( - \frac{{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{x}cos(Pix)}{(-{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}cos(Pix) + (\frac{1}{2}sqrt(5) + \frac{1}{2})^{x})} + \frac{(\frac{1}{2}sqrt(5) + \frac{1}{2})^{x}}{(-{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}cos(Pix) + (\frac{1}{2}sqrt(5) + \frac{1}{2})^{x})}\right)}{dx}\\=& - (\frac{-(-({\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}((1 + 0)ln(\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}) + \frac{(x + 2)((\frac{-(\frac{1}{2}*0*\frac{1}{2}*5^{\frac{1}{2}} + 0)}{(\frac{1}{2}sqrt(5) + \frac{1}{2})^{2}}))}{(\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})})}))cos(Pix) - {\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}*-sin(Pix)Pi + ((\frac{1}{2}sqrt(5) + \frac{1}{2})^{x}((1)ln(\frac{1}{2}sqrt(5) + \frac{1}{2}) + \frac{(x)(\frac{1}{2}*0*\frac{1}{2}*5^{\frac{1}{2}} + 0)}{(\frac{1}{2}sqrt(5) + \frac{1}{2})})))}{(-{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}cos(Pix) + (\frac{1}{2}sqrt(5) + \frac{1}{2})^{x})^{2}}){\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{x}cos(Pix) - \frac{({\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{x}((1)ln(\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}) + \frac{(x)((\frac{-(\frac{1}{2}*0*\frac{1}{2}*5^{\frac{1}{2}} + 0)}{(\frac{1}{2}sqrt(5) + \frac{1}{2})^{2}}))}{(\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})})}))cos(Pix)}{(-{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}cos(Pix) + (\frac{1}{2}sqrt(5) + \frac{1}{2})^{x})} - \frac{{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{x}*-sin(Pix)Pi}{(-{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}cos(Pix) + (\frac{1}{2}sqrt(5) + \frac{1}{2})^{x})} + (\frac{-(-({\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}((1 + 0)ln(\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}) + \frac{(x + 2)((\frac{-(\frac{1}{2}*0*\frac{1}{2}*5^{\frac{1}{2}} + 0)}{(\frac{1}{2}sqrt(5) + \frac{1}{2})^{2}}))}{(\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})})}))cos(Pix) - {\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}*-sin(Pix)Pi + ((\frac{1}{2}sqrt(5) + \frac{1}{2})^{x}((1)ln(\frac{1}{2}sqrt(5) + \frac{1}{2}) + \frac{(x)(\frac{1}{2}*0*\frac{1}{2}*5^{\frac{1}{2}} + 0)}{(\frac{1}{2}sqrt(5) + \frac{1}{2})})))}{(-{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}cos(Pix) + (\frac{1}{2}sqrt(5) + \frac{1}{2})^{x})^{2}})(\frac{1}{2}sqrt(5) + \frac{1}{2})^{x} + \frac{((\frac{1}{2}sqrt(5) + \frac{1}{2})^{x}((1)ln(\frac{1}{2}sqrt(5) + \frac{1}{2}) + \frac{(x)(\frac{1}{2}*0*\frac{1}{2}*5^{\frac{1}{2}} + 0)}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}))}{(-{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}cos(Pix) + (\frac{1}{2}sqrt(5) + \frac{1}{2})^{x})}\\=&\frac{-{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(2x + 4)}ln(\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})})cos^{2}(Pix)}{(-{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}cos(Pix) + (\frac{1}{2}sqrt(5) + \frac{1}{2})^{x})^{2}} + \frac{Pi{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(2x + 4)}sin(Pix)cos(Pix)}{(-{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}cos(Pix) + (\frac{1}{2}sqrt(5) + \frac{1}{2})^{x})^{2}} + \frac{(\frac{1}{2}sqrt(5) + \frac{1}{2})^{x}{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{x}ln(\frac{1}{2}sqrt(5) + \frac{1}{2})cos(Pix)}{(-{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}cos(Pix) + (\frac{1}{2}sqrt(5) + \frac{1}{2})^{x})^{2}} - \frac{{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{x}ln(\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})})cos(Pix)}{(-{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}cos(Pix) + (\frac{1}{2}sqrt(5) + \frac{1}{2})^{x})} + \frac{Pi{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{x}sin(Pix)}{(-{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}cos(Pix) + (\frac{1}{2}sqrt(5) + \frac{1}{2})^{x})} + \frac{{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}(\frac{1}{2}sqrt(5) + \frac{1}{2})^{x}ln(\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})})cos(Pix)}{(-{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}cos(Pix) + (\frac{1}{2}sqrt(5) + \frac{1}{2})^{x})^{2}} - \frac{Pi{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}(\frac{1}{2}sqrt(5) + \frac{1}{2})^{x}sin(Pix)}{(-{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}cos(Pix) + (\frac{1}{2}sqrt(5) + \frac{1}{2})^{x})^{2}} - \frac{(\frac{1}{2}sqrt(5) + \frac{1}{2})^{(2(x))}ln(\frac{1}{2}sqrt(5) + \frac{1}{2})}{(-{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}cos(Pix) + (\frac{1}{2}sqrt(5) + \frac{1}{2})^{x})^{2}} + \frac{(\frac{1}{2}sqrt(5) + \frac{1}{2})^{x}ln(\frac{1}{2}sqrt(5) + \frac{1}{2})}{(-{\frac{1}{(\frac{1}{2}sqrt(5) + \frac{1}{2})}}^{(x + 2)}cos(Pix) + (\frac{1}{2}sqrt(5) + \frac{1}{2})^{x})}\\ \end{split}\end{equation} \]Your problem has not been solved here? 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