(\partial)/(\partial x)(x^3e^{sqrt(4xy)})

Lösung

\frac{\partial}{\partial x} (x^{3} e^{4 x y})

e^{2 x y} x^{\frac{5}{2}} y + 3 e^{2 x y} x^{2}

Schritte zur Lösung

\frac{\partial}{\partial x} (x^{3} e^{4 x y})

Vereinfache

x^{3} e^{4 x y} : e^{2 x y} x^{3}

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

Behandle

y

als Konstante

Wende die Produktregel an:

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

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

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

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

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

= \frac{y e ^{2 y x}}{x} x^{3} + 3 x^{2} e^{2 x y}

\frac{y e ^{2 y x}}{x} x^{3} = e^{2 x y} x^{\frac{5}{2}} y

= e^{2 x y} x^{\frac{5}{2}} y + 3 e^{2 x y} x^{2}

y^{^{'}} = \frac{y}{t} - \frac{1}{\frac{y}{t}}

t an g e n t f (x) = x + \frac{2}{x}, at x = 2

\int \frac{50}{x ^{0.25}} d x

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