derivative y=(x^{10}-5x^6-12x)/(x^6)

Lösung

ableitung von

y = \frac{x ^{10} - 5 x ^{6} - 12 x}{x ^{6}}

\frac{4 ( x ^{9} + 15 )}{x ^{6}}

Schritte zur Lösung

\frac{d}{d x} (\frac{x ^{10} - 5 x ^{6} - 12 x}{x ^{6}})

Wende die Quotientenregel an:

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

= \frac{\frac{d}{d x} ( x ^{10} - 5 x ^{6} - 12 x ) x ^{6} - \frac{d}{d x} ( x ^{6} ) ( x ^{10} - 5 x ^{6} - 12 x )}{( x ^{6} ) ^{2}}

\frac{d}{d x} (x^{10} - 5 x^{6} - 12 x) = 10 x^{9} - 30 x^{5} - 12

\frac{d}{d x} (x^{6}) = 6 x^{5}

= \frac{( 10 x ^{9} - 30 x ^{5} - 12 ) x ^{6} - 6 x ^{5} ( x ^{10} - 5 x ^{6} - 12 x )}{( x ^{6} ) ^{2}}

Vereinfache

\frac{( 10 x ^{9} - 30 x ^{5} - 12 ) x ^{6} - 6 x ^{5} ( x ^{10} - 5 x ^{6} - 12 x )}{( x ^{6} ) ^{2}} : \frac{4 ( x ^{9} + 15 )}{x ^{6}}

= \frac{4 ( x ^{9} + 15 )}{x ^{6}}

\frac{d y}{d x} - \frac{9}{x} y = \frac{y ^{3}}{x ^{13}}, y (1) = 1

\frac{\partial}{\partial x} (r^{2} + s^{2})

\frac{\partial}{\partial x} (e^{3 x} d y)

\frac{\partial}{\partial x} (x^{2} + 6 x y + y^{2})

(x^{2} + 1) \frac{d y}{d x} + 3 x (y - 1) = 0, y (0) = 9