Question-94767

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

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

y (x) = 5^{- x} + 5 \arcsin (\frac{x}{6}) + \ln (x^{2} + 4) + e^{\frac{- 7}{2} x} + 3^{- x} \ln 5

\Rightarrow y (x) = u (x) + v (x) with u (x) = 5^{- x} + 3^{- x} \ln (5)

and v (x) = 5 \arcsin (\frac{x}{6}) + \ln (x^{2} + 4) + e^{- \frac{7 x}{2}}

we have u (x) = e^{- xln 5} + e^{- xln (3)} \ln 5 \Rightarrow u^{^{'}} (x) = - \ln 5 e^{- xln 5}

- \ln 3 e^{- xln 3} \ln 5 = - \ln 5 \times 5^{- x} - \ln 3 . \ln 5 3^{- x}

v^{^{'}} (x) = \frac{5}{6} \times \frac{1}{\sqrt{1 - \frac{x^{2}}{36}}} + \frac{2 x}{x^{2} + 4} - \frac{7}{2} e^{- \frac{7 x}{2}} \Rightarrow

y^{^{'}} (x) = - \ln 5 \times 5^{- x} - \ln 3 . \ln 5 \times 3^{- x} + \frac{5}{6 \sqrt{1 - \frac{x^{2}}{36}}} + \frac{2 x}{x^{2} + 4} - \frac{7}{2} e^{- \frac{7 x}{2}}