derivative of (tan(x)/(x^3))

Lösung

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

\frac{x sec ^{2} ( x ) - 3 tan ( x )}{x ^{4}}

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

= \frac{\frac{d}{d x} ( tan ( x ) ) x ^{3} - \frac{d}{d x} ( x ^{3} ) tan ( x )}{( x ^{3} ) ^{2}}

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

\frac{d}{d x} (x^{3}) = 3 x^{2}

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

Vereinfache

\frac{sec ^{2} ( x ) x ^{3} - 3 x ^{2} tan ( x )}{( x ^{3} ) ^{2}} : \frac{x sec ^{2} ( x ) - 3 tan ( x )}{x ^{4}}

= \frac{x sec ^{2} ( x ) - 3 tan ( x )}{x ^{4}}

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