derivative of y/(x^2+y^2+1)

Lösung

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

Multipliziere aus

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

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

\frac{d}{d x} (ln (x) sec (x))

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

\int x^{6} cos (x) d x

d er i v a t i v e ln (cos (5 x))

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