derivative of (tan(x)/(6xsec(x)))

Lösung

\frac{d}{d x} (\frac{tan ( x )}{6 x sec ( x )})

\frac{x sec ^{3} ( x ) - tan ( x ) ( sec ( x ) + x sec ( x ) tan ( x ) )}{6 x ^{2} sec ^{2} ( x )}

Schritte zur Lösung

\frac{d}{d x} (\frac{tan ( x )}{6 x sec ( x )})

Entferne die Konstante:

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

= \frac{1}{6} \frac{d}{d x} (\frac{tan ( x )}{x sec ( x )})

Wende die Quotientenregel an:

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

= \frac{1}{6} \cdot \frac{\frac{d}{d x} ( tan ( x ) ) x sec ( x ) - \frac{d}{d x} ( x sec ( x ) ) tan ( x )}{( x sec ( x ) ) ^{2}}

\frac{d}{d x} (tan (x)) = sec^{2} (x)

\frac{d}{d x} (x sec (x)) = sec (x) + x sec (x) tan (x)

= \frac{1}{6} \cdot \frac{sec ^{2} ( x ) x sec ( x ) - ( sec ( x ) + x sec ( x ) tan ( x ) ) tan ( x )}{( x sec ( x ) ) ^{2}}

Vereinfache

\frac{1}{6} \cdot \frac{sec ^{2} ( x ) x sec ( x ) - ( sec ( x ) + x sec ( x ) tan ( x ) ) tan ( x )}{( x sec ( x ) ) ^{2}} : \frac{x sec ^{3} ( x ) - tan ( x ) ( sec ( x ) + x sec ( x ) tan ( x ) )}{6 x ^{2} sec ^{2} ( x )}

= \frac{x sec ^{3} ( x ) - tan ( x ) ( sec ( x ) + x sec ( x ) tan ( x ) )}{6 x ^{2} sec ^{2} ( x )}