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

Lösung

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

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

Schritte zur Lösung

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

Behandle

y, z

als Konstanten

Wende die Quotientenregel an:

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

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

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

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

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

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

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

y^{^{''}} - x y^{^{'}} + y = 0

\int (x^{\frac{1}{3}}) d x

d er i v a t i v e f (x) = x (x + 3)^{3}

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

x \to 0 lim (\frac{e x}{x})