d/(d{x)}(({x})/(sqrt({x)^2+{y)^2+{z}^2}})

Lösung

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

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

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

in v erse l a pl a ce \frac{s + 1}{( s ^{2} - 2 s + 5 )}

\frac{\partial}{\partial x} (x yz + x y^{5} + y z^{5} + z x^{5})

a re a y = x^{2} - 3, y = 1

\int 3 e^{4 x} + 2 e^{4 x} d x

\frac{d}{d x} (x^{3} - 12 x y + 8 y^{3})