derivative of (1+csc(x^7)/(1-cot(x^7)))

Lösung

\frac{d}{d x} (\frac{1 + csc ( x ^{7} )}{1 - cot ( x ^{7} )})

\frac{- 7 x ^{6} cot ( x ^{7} ) csc ( x ^{7} ) ( 1 - cot ( x ^{7} ) ) - 7 x ^{6} csc ^{2} ( x ^{7} ) ( 1 + csc ( x ^{7} ) )}{( 1 - cot ( x ^{7} ) ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{1 + csc ( x ^{7} )}{1 - cot ( x ^{7} )})

Wende die Quotientenregel an:

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

= \frac{\frac{d}{d x} ( 1 + csc ( x ^{7} ) ) ( 1 - cot ( x ^{7} ) ) - \frac{d}{d x} ( 1 - cot ( x ^{7} ) ) ( 1 + csc ( x ^{7} ) )}{( 1 - cot ( x ^{7} ) ) ^{2}}

\frac{d}{d x} (1 + csc (x^{7})) = - 7 x^{6} cot (x^{7}) csc (x^{7})

\frac{d}{d x} (1 - cot (x^{7})) = 7 x^{6} csc^{2} (x^{7})

= \frac{( - 7 x ^{6} cot ( x ^{7} ) csc ( x ^{7} ) ) ( 1 - cot ( x ^{7} ) ) - 7 x ^{6} csc ^{2} ( x ^{7} ) ( 1 + csc ( x ^{7} ) )}{( 1 - cot ( x ^{7} ) ) ^{2}}

Vereinfache

= \frac{- 7 x ^{6} cot ( x ^{7} ) csc ( x ^{7} ) ( 1 - cot ( x ^{7} ) ) - 7 x ^{6} csc ^{2} ( x ^{7} ) ( 1 + csc ( x ^{7} ) )}{( 1 - cot ( x ^{7} ) ) ^{2}}