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

Lösung

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

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

Schritte zur Lösung

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

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

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

Vereinfache

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

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

x \to - 2 + lim (\frac{1}{x ^{2} - 4})

t an g e n t f (x) = x + 8, (1, 3)

n = 1 \sum \infty 3^{n} 5^{1 - n}

x \to \infty lim (\frac{x ^{\frac{1}{2}}}{x ^{\frac{2}{3}}})

x \to 0 + lim (\frac{2}{e ^{x} + 1})