derivative of (xy/(x^2-y^2))

Lösung

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Multipliziere aus

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

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

cr i t i c a l 3 x^{2} \cdot (x - 2)^{2}

cr i t i c a l f (x) = x^{3} - 9 x^{2} + 24 x - 10

cr i t i c a l f (x) = x^{4} - 4 x^{3} + 4 x^{2}

cr i t i c a l f (x) = \frac{1}{4} x^{4} - 2 x^{2}

cr i t i c a l f (x) = x^{3} + 9 x^{2} - 81 x