derivative of (-9x^2+4y^2-12xy/(4y^2-9x^2))

Lösung

\frac{d}{d x} (\frac{- 9 x ^{2} + 4 y ^{2} - 12 x y}{4 y ^{2} - 9 x ^{2}})

\frac{108 x ^{3} \frac{d y}{d x} - 108 x ^{2} y + 48 x y ^{2} \frac{d y}{d x} - 48 y ^{3}}{( 4 y ^{2} - 9 x ^{2} ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{- 9 x ^{2} + 4 y ^{2} - 12 x y}{4 y ^{2} - 9 x ^{2}})

Wende die Quotientenregel an:

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

= \frac{\frac{d}{d x} ( - 9 x ^{2} + 4 y ^{2} - 12 x y ) ( 4 y ^{2} - 9 x ^{2} ) - \frac{d}{d x} ( 4 y ^{2} - 9 x ^{2} ) ( - 9 x ^{2} + 4 y ^{2} - 12 x y )}{( 4 y ^{2} - 9 x ^{2} ) ^{2}}

\frac{d}{d x} (- 9 x^{2} + 4 y^{2} - 12 x y) = - 18 x + 8 y \frac{d y}{d x} - 12 (y + x \frac{d y}{d x})

\frac{d}{d x} (4 y^{2} - 9 x^{2}) = 8 y \frac{d y}{d x} - 18 x

= \frac{( - 18 x + 8 y \frac{d y}{d x} - 12 ( y + x \frac{d y}{d x} ) ) ( 4 y ^{2} - 9 x ^{2} ) - ( 8 y \frac{d y}{d x} - 18 x ) ( - 9 x ^{2} + 4 y ^{2} - 12 x y )}{( 4 y ^{2} - 9 x ^{2} ) ^{2}}

Multipliziere aus

(- 18 x + 8 y \frac{d y}{d x} - 12 (y + x \frac{d y}{d x})) (4 y^{2} - 9 x^{2}) - (8 y \frac{d y}{d x} - 18 x) (- 9 x^{2} + 4 y^{2} - 12 x y) : 108 x^{3} \frac{d y}{d x} - 108 x^{2} y + 48 x y^{2} \frac{d y}{d x} - 48 y^{3}

= \frac{108 x ^{3} \frac{d y}{d x} - 108 x ^{2} y + 48 x y ^{2} \frac{d y}{d x} - 48 y ^{3}}{( 4 y ^{2} - 9 x ^{2} ) ^{2}}

cr i t i c a l \frac{x ^{3}}{3 x ^{2} + 1}

cr i t i c a l \frac{1}{3} x^{3} - 9 x + 2

cr i t i c a l f (x) = x^{3} - 9 x^{2} + 4

cr i t i c a l e^{x^{2} - 4 x - 1}

cr i t i c a l f (x) = \frac{1}{3} x^{3} + x^{2} - 3 x - 8