derivative of y/(x+y^2)

Lösung

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

Multipliziere aus

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

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

\int \frac{x ^{2} - 2 x - 1}{2 x ^{3} + 3 x ^{2} - 2 x} d x

\int \frac{x ^{2}}{e ^{5 x}} d x

\frac{d}{d x} (3 x^{2} cos (x) cot (x))

x \to - \infty lim (\frac{5 x ^{2} - sin ( 3 x )}{x ^{2} + 10})

\frac{d}{d x} (tan (5 - sin (2 x)))