(\partial}{\partial x}((e^{\frac{x^2)/(4t)}tx)/2)

Lösung

\frac{\partial}{\partial x} (\frac{e ^{\frac{x ^{2}}{4 t}} t x}{2})

\frac{e ^{\frac{x ^{2}}{4 t}} x ^{2} + 2 e ^{\frac{x ^{2}}{4 t}} t}{4}

Schritte zur Lösung

\frac{\partial}{\partial x} (\frac{e ^{\frac{x ^{2}}{4 t}} t x}{2})

Behandle

t

als Konstante

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{t}{2} \frac{\partial}{\partial x} (e^{\frac{x ^{2}}{4 t}} x)

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = e^{\frac{x ^{2}}{4 t}}, g = x

= \frac{t}{2} (\frac{\partial}{\partial x} (e^{\frac{x ^{2}}{4 t}}) x + \frac{\partial}{\partial x} (x) e^{\frac{x ^{2}}{4 t}})

\frac{\partial}{\partial x} (e^{\frac{x ^{2}}{4 t}}) = \frac{e ^{\frac{x ^{2}}{4 t}} x}{2 t}

\frac{\partial}{\partial x} (x) = 1

= \frac{t}{2} (\frac{e ^{\frac{x ^{2}}{4 t}} x}{2 t} x + 1 \cdot e^{\frac{x ^{2}}{4 t}})

Vereinfache

\frac{t}{2} (\frac{e ^{\frac{x ^{2}}{4 t}} x}{2 t} x + 1 \cdot e^{\frac{x ^{2}}{4 t}}) : \frac{e ^{\frac{x ^{2}}{4 t}} x ^{2} + 2 e ^{\frac{x ^{2}}{4 t}} t}{4}

= \frac{e ^{\frac{x ^{2}}{4 t}} x ^{2} + 2 e ^{\frac{x ^{2}}{4 t}} t}{4}

\int \frac{1}{x ^{2}} sin (\frac{1}{x}) d x

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y^{^{''}} + 4 y^{^{'}} + 4 y = 0