derivative of (3+xy/(sqrt(x)))

Lösung

\frac{d}{d x} (\frac{3 + x y}{x})

\frac{2 x ^{2} \frac{d y}{d x} + x y - 3}{2 x x}

Schritte zur Lösung

\frac{d}{d x} (\frac{3 + x y}{x})

Wende die Quotientenregel an:

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

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

\frac{d}{d x} (3 + x y) = x \frac{d y}{d x} + y

\frac{d}{d x} (x) = \frac{1}{2 x}

= \frac{( x \frac{d y}{d x} + y ) x - \frac{1}{2 x} ( 3 + x y )}{( x ) ^{2}}

Vereinfache

\frac{( x \frac{d y}{d x} + y ) x - \frac{1}{2 x} ( 3 + x y )}{( x ) ^{2}} : \frac{2 x ^{2} \frac{d y}{d x} + x y - 3}{2 x x}

= \frac{2 x ^{2} \frac{d y}{d x} + x y - 3}{2 x x}

\frac{d}{d x} (2 3 x^{2})

\int 3 x^{7} d x

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