derivative of (6x/(5-cot(x)))

Lösung

\frac{d}{d x} (\frac{6 x}{5 - cot ( x )})

\frac{6 ( 5 - cot ( x ) - x csc ^{2} ( x ) )}{( 5 - cot ( x ) ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{6 x}{5 - cot ( x )})

Entferne die Konstante:

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

= 6 \frac{d}{d x} (\frac{x}{5 - cot ( x )})

Wende die Quotientenregel an:

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

= 6 \cdot \frac{\frac{d x}{d x} ( 5 - cot ( x ) ) - \frac{d}{d x} ( 5 - cot ( x ) ) x}{( 5 - cot ( x ) ) ^{2}}

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

\frac{d}{d x} (5 - cot (x)) = csc^{2} (x)

= 6 \cdot \frac{1 \cdot ( 5 - cot ( x ) ) - csc ^{2} ( x ) x}{( 5 - cot ( x ) ) ^{2}}

Vereinfache

6 \cdot \frac{1 \cdot ( 5 - cot ( x ) ) - csc ^{2} ( x ) x}{( 5 - cot ( x ) ) ^{2}} : \frac{6 ( 5 - cot ( x ) - x csc ^{2} ( x ) )}{( 5 - cot ( x ) ) ^{2}}

= \frac{6 ( 5 - cot ( x ) - x csc ^{2} ( x ) )}{( 5 - cot ( x ) ) ^{2}}