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

Lösung

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

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

Schritte zur Lösung

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

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

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

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

Vereinfache

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

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

x d x - y^{3} d y = 0

\frac{d}{d t} (- A e^{- t})

\int_{1}^{t} - t + 2 d t

l a pl a ce t r an s f or m 12 t e^{10 t}

\frac{d y}{d x} - 2 = - y