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