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

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die Summen-/Differenzregel an:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

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

\frac{d}{d x} (y z) = 0

= z (\frac{1 + \frac{d}{d x} ( z )}{x + z} - 0)

Vereinfache

z (\frac{1 + \frac{d}{d x} ( z )}{x + z} - 0) : \frac{( \frac{d}{d x} ( z ) + 1 ) z}{x + z}

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

e x p an d 3 (n - 1 = 5 n + 3 - 2 n)

e x p an d \frac{1}{( 5 x - 2 ) ^{2}}

e x p an d \frac{( x - 3 ) ( x + 1 )}{( x + 2 )}

e x p an d cos (\frac{π}{2} (1 - a))

e x p an d C (500 = 8000 + C)