(\partial)/(\partial x)(txe^{(x^2)/(2t)})

Lösung

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

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

Schritte zur Lösung

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

Behandle

t

als Konstante

Entferne die Konstante:

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

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

Wende die Produktregel an:

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

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

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

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

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

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

Vereinfache

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

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

a re a x^{4} - 9 x^{2}, x^{2} - 9

d er i v a t i v e 2 csc (x)

\frac{\partial}{\partial x} (\frac{1}{2} ln (x^{2} + y^{2}))

t an g e n t y = x^{4} + 4 e^{x}

\int (2 x sin (y) - y sin (x)) d x