derivative of (cot(x)/(1+csc(x)))

Lösung

\frac{d}{d x} (\frac{cot ( x )}{1 + csc ( x )})

\frac{- csc ^{2} ( x ) ( 1 + csc ( x ) ) + cot ^{2} ( x ) csc ( x )}{( 1 + csc ( x ) ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{cot ( x )}{1 + csc ( x )})

Wende die Quotientenregel an:

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

= \frac{\frac{d}{d x} ( cot ( x ) ) ( 1 + csc ( x ) ) - \frac{d}{d x} ( 1 + csc ( x ) ) cot ( x )}{( 1 + csc ( x ) ) ^{2}}

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

\frac{d}{d x} (1 + csc (x)) = - cot (x) csc (x)

= \frac{( - csc ^{2} ( x ) ) ( 1 + csc ( x ) ) - ( - cot ( x ) csc ( x ) ) cot ( x )}{( 1 + csc ( x ) ) ^{2}}

Vereinfache

\frac{( - csc ^{2} ( x ) ) ( 1 + csc ( x ) ) - ( - cot ( x ) csc ( x ) ) cot ( x )}{( 1 + csc ( x ) ) ^{2}} : \frac{- csc ^{2} ( x ) ( 1 + csc ( x ) ) + cot ^{2} ( x ) csc ( x )}{( 1 + csc ( x ) ) ^{2}}

= \frac{- csc ^{2} ( x ) ( 1 + csc ( x ) ) + cot ^{2} ( x ) csc ( x )}{( 1 + csc ( x ) ) ^{2}}

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

c^{^{'}}

\frac{d ^{4}}{d x ^{4}} (3 x^{4})

d er i v a t i v e \frac{4}{x + 1}

\int (12 x + 2) d x