d/(d{y)}(({y})/({x)+{y}+{z}})

Lösung

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

\frac{d}{d y} (x + y + z) = \frac{d}{d y} (x) + 1 + \frac{d}{d y} (z)

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

Vereinfache

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

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

d er i v a t i v e 7 x - 2 x

t an g e n t y = x^{2} + 9, (4, 25)

\frac{d}{d x} (arctan (x^{3} - x))

n or ma l cos (t) + t sin (t), at t = π

a re a y = x, y = \frac{1}{4} x + 1, x = 0