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Question Number 94767 by Hamida last updated on 20/May/20

Answered by mathmax by abdo last updated on 21/May/20

$$\mathrm{y}\left(\mathrm{x}\right)=5^{-\mathrm{x}}+5\arcsin \left(\frac{\mathrm{x}}{6}\right)+\ln ({\mathrm{x}}^2+4)+{\mathrm{e}}^{\frac{-7}{2}\mathrm{x}}+3^{-\mathrm{x}}\ln 5$$$$\Rightarrow \mathrm{y}\left(\mathrm{x}\right)=\mathrm{u}\left(\mathrm{x}\right)+\mathrm{v}\left(\mathrm{x}\right)\mathrm{with}\mathrm{u}\left(\mathrm{x}\right)=5^{-\mathrm{x}}+3^{-\mathrm{x}}\ln \left(5\right)$$$$\mathrm{and}\mathrm{v}\left(\mathrm{x}\right)=5\arcsin \left(\frac{\mathrm{x}}{6}\right)+\ln ({\mathrm{x}}^2+4)+{\mathrm{e}}^{-\frac{7\mathrm{x}}{2}}$$$$\mathrm{we}\mathrm{have}\mathrm{u}\left(\mathrm{x}\right)={\mathrm{e}}^{-\mathrm{xln}5}+{\mathrm{e}}^{-\mathrm{xln}\left(3\right)}\ln 5\Rightarrow {\mathrm{u}}^{{}^{\prime }}\left(\mathrm{x}\right)=-\ln 5{\mathrm{e}}^{-\mathrm{xln}5}$$$$-\ln 3{\mathrm{e}}^{-\mathrm{xln}3}\ln 5=-\ln 5\times 5^{-\mathrm{x}}-\ln 3.\ln 53^{-\mathrm{x}}$$$${\mathrm{v}}^{{}^{\prime }}\left(\mathrm{x}\right)=\frac{5}{6}\times \frac{1}{\sqrt{1-\frac{{\mathrm{x}}^2}{36}}}+\frac{2\mathrm{x}}{{\mathrm{x}}^2+4}-\frac{7}{2}{\mathrm{e}}^{-\frac{7\mathrm{x}}{2}}\Rightarrow$$$${\mathrm{y}}^{{}^{\prime }}\left(\mathrm{x}\right)=-\ln 5\times 5^{-\mathrm{x}}-\ln 3.\ln 5\times 3^{-\mathrm{x}}+\frac{5}{6\sqrt{1-\frac{{\mathrm{x}}^2}{36}}}+\frac{2\mathrm{x}}{{\mathrm{x}}^2+4}-\frac{7}{2}{\mathrm{e}}^{-\frac{7\mathrm{x}}{2}}$$$$$$

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