derivative of x^3e^{1/3 x}+(2x/(5+x))

Lösung

\frac{d}{d x} (x^{3} e^{\frac{1}{3} x} + \frac{2 x}{5 + x})

\frac{e ^{\frac{x}{3}} x ^{5} + 19 e ^{\frac{x}{3}} x ^{4} + 115 e ^{\frac{x}{3}} x ^{3} + 225 e ^{\frac{x}{3}} x ^{2} + 30}{3 ( x + 5 ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (x^{3} e^{\frac{1}{3} x} + \frac{2 x}{5 + x})

Vereinfache

x^{3} e^{\frac{1}{3} x} + \frac{2 x}{5 + x} : e^{\frac{x}{3}} x^{3} + \frac{2 x}{5 + x}

= \frac{d}{d x} (e^{\frac{x}{3}} x^{3} + \frac{2 x}{5 + x})

Wende die Summen-/Differenzregel an:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (e^{\frac{x}{3}} x^{3}) + \frac{d}{d x} (\frac{2 x}{5 + x})

\frac{d}{d x} (e^{\frac{x}{3}} x^{3}) = \frac{e ^{\frac{x}{3}} x ^{3}}{3} + 3 e^{\frac{x}{3}} x^{2}

\frac{d}{d x} (\frac{2 x}{5 + x}) = \frac{10}{( 5 + x ) ^{2}}

= \frac{e ^{\frac{x}{3}} x ^{3}}{3} + 3 e^{\frac{x}{3}} x^{2} + \frac{10}{( 5 + x ) ^{2}}

Vereinfache

\frac{e ^{\frac{x}{3}} x ^{3}}{3} + 3 e^{\frac{x}{3}} x^{2} + \frac{10}{( 5 + x ) ^{2}} : \frac{e ^{\frac{x}{3}} x ^{5} + 19 e ^{\frac{x}{3}} x ^{4} + 115 e ^{\frac{x}{3}} x ^{3} + 225 e ^{\frac{x}{3}} x ^{2} + 30}{3 ( x + 5 ) ^{2}}

= \frac{e ^{\frac{x}{3}} x ^{5} + 19 e ^{\frac{x}{3}} x ^{4} + 115 e ^{\frac{x}{3}} x ^{3} + 225 e ^{\frac{x}{3}} x ^{2} + 30}{3 ( x + 5 ) ^{2}}

\int \frac{1}{1 - 4 x - x ^{2}} d x

\frac{d}{d x} (x^{\frac{7}{8}})

\int (csc^{2} (x) - a^{x}) d x

x \to \infty lim (0^{- x})

y^{^{''}} - 2 y^{^{'}} + 1 y = 0, y (0) = 5, y^{^{'}} (0) = 4