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

Lösung

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

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

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

\int \frac{x}{- x ^{2} + 4 x + 4} d x

x \to \infty lim (\frac{4 ^{x + 1} + 2 ^{x}}{4 ^{x}})

\frac{d}{d x} (- 7 ln (5 x^{2} - 11))

\int 1 0^{x^{2}} x d x

x \to - \infty lim (x^{3} e^{x})