(\partial)/(\partial x)(xy^2e^{3x^2y-y^2})

Lösung

\frac{\partial}{\partial x} (x y^{2} e^{3 x^{2} y - y^{2}})

y^{2} (e^{3 x^{2} y - y^{2}} + 6 e^{3 x^{2} y - y^{2}} x^{2} y)

Schritte zur Lösung

\frac{\partial}{\partial x} (x y^{2} e^{3 x^{2} y - y^{2}})

Behandle

y

als Konstante

Entferne die Konstante:

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

= y^{2} \frac{\partial}{\partial x} (x e^{3 x^{2} y - y^{2}})

Wende die Produktregel an:

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

f = x, g = e^{3 x^{2} y - y^{2}}

= y^{2} (\frac{\partial}{\partial x} (x) e^{3 x^{2} y - y^{2}} + \frac{\partial}{\partial x} (e^{3 x^{2} y - y^{2}}) x)

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

\frac{\partial}{\partial x} (e^{3 x^{2} y - y^{2}}) = e^{3 x^{2} y - y^{2}} \cdot 6 y x

= y^{2} (1 \cdot e^{3 x^{2} y - y^{2}} + e^{3 x^{2} y - y^{2}} \cdot 6 y xx)

Vereinfache

y^{2} (1 \cdot e^{3 x^{2} y - y^{2}} + e^{3 x^{2} y - y^{2}} \cdot 6 y xx) : y^{2} (e^{3 x^{2} y - y^{2}} + 6 e^{3 x^{2} y - y^{2}} x^{2} y)

= y^{2} (e^{3 x^{2} y - y^{2}} + 6 e^{3 x^{2} y - y^{2}} x^{2} y)

d er i v a t i v e \frac{( x + x )}{x ^{2}}

in v erse l a pl a ce \frac{8}{s - 1} - \frac{3}{s}

x \to 0 lim (\frac{x}{x ^{2} + 4 x})

n = 1 \sum \infty (- 3)^{n} x^{n}

x \to 4 - lim (\frac{∣ 2 x ∣}{x})