(\partial)/(\partial x)(15xye^{-x^2-y^2})

Lösung

\frac{\partial}{\partial x} (15 x y e^{- x^{2} - y^{2}})

15 y (e^{- x^{2} - y^{2}} - 2 e^{- x^{2} - y^{2}} x^{2})

Schritte zur Lösung

\frac{\partial}{\partial x} (15 x y e^{- x^{2} - y^{2}})

Behandle

y

als Konstante

Entferne die Konstante:

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

= 15 y \frac{\partial}{\partial x} (x e^{- x^{2} - y^{2}})

Wende die Produktregel an:

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

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

= 15 y (\frac{\partial}{\partial x} (x) e^{- x^{2} - y^{2}} + \frac{\partial}{\partial x} (e^{- x^{2} - y^{2}}) x)

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

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

= 15 y (1 \cdot e^{- x^{2} - y^{2}} + (- 2 e^{- x^{2} - y^{2}} x) x)

Vereinfache

15 y (1 \cdot e^{- x^{2} - y^{2}} + (- 2 e^{- x^{2} - y^{2}} x) x) : 15 y (e^{- x^{2} - y^{2}} - 2 e^{- x^{2} - y^{2}} x^{2})

= 15 y (e^{- x^{2} - y^{2}} - 2 e^{- x^{2} - y^{2}} x^{2})

\int \frac{x}{( 16 + x ^{2} ) ^{\frac{3}{2}}} d x

p a r i t y y = ln (\frac{1 + sin ( x )}{1 - sin ( x )})

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

x \to 1 lim (\frac{sin ( x - 1 )}{x})

\int \frac{ln ( 2 x )}{3 x} d x