(\partial)/(\partial x)((xy)/(sqrt(x^2+9)))

Lösung

\frac{\partial}{\partial x} (\frac{x y}{x ^{2} + 9})

\frac{9 y}{( x ^{2} + 9 ) x ^{2} + 9}

Schritte zur Lösung

\frac{\partial}{\partial x} (\frac{x y}{x ^{2} + 9})

Behandle

y

als Konstante

Entferne die Konstante:

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

= y \frac{\partial}{\partial x} (\frac{x}{x ^{2} + 9})

Wende die Quotientenregel an:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= y \frac{\frac{\partial}{\partial x} ( x ) x ^{2} + 9 - \frac{\partial}{\partial x} ( x ^{2} + 9 ) x}{( x ^{2} + 9 ) ^{2}}

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

\frac{\partial}{\partial x} (x^{2} + 9) = \frac{x}{x ^{2} + 9}

= y \frac{1 \cdot x ^{2} + 9 - \frac{x}{x ^{2} + 9} x}{( x ^{2} + 9 ) ^{2}}

Vereinfache

y \frac{1 \cdot x ^{2} + 9 - \frac{x}{x ^{2} + 9} x}{( x ^{2} + 9 ) ^{2}} : \frac{9 y}{( x ^{2} + 9 ) x ^{2} + 9}

= \frac{9 y}{( x ^{2} + 9 ) x ^{2} + 9}

\int 8 36 - 4 x - 32 d x

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