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

Lösung

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

\frac{d x}{d x} = 1

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

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

Vereinfache

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

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

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

\int_{0}^{1} - 8 x 5^{x} d x

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

\int cos^{8} (θ) sin (θ) d θ

s l o p e y - 3 = 5 (x - 2) y - 3 = 5 (x - 2)