(\partial)/(\partial x)((2x)/((x^2+y^2)))

Lösung

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

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

Schritte zur Lösung

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

Behandle

y

als Konstante

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

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

Wende die Quotientenregel an:

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

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

\frac{\partial}{\partial x} (x) = 1

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

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

Vereinfache

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

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

d er i v a t i v e (2 x + 5) (5 x^{2} + 1)

\int_{0}^{\frac{3 π}{2}} 7 ∣ sin (x) ∣ d x

a re a f (x) = e^{x}, g (x) = cos (x), x = 0, x = \frac{π}{2}

a re a y = x^{2} - 3 x, y = x

\int_{0}^{2} x (1 - x) d x