derivative of (tan(20x)/(x^{2/3)+3x})

Lösung

\frac{d}{d x} (\frac{tan ( 20 x )}{x ^{\frac{2}{3}} + 3 x})

\frac{- 2 tan ( 20 x ) + 60 x sec ^{2} ( 20 x ) + 180 x ^{\frac{4}{3}} sec ^{2} ( 20 x ) - 9 x ^{\frac{1}{3}} tan ( 20 x )}{3 x ^{\frac{1}{3}} ( x ^{\frac{2}{3}} + 3 x ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{tan ( 20 x )}{x ^{\frac{2}{3}} + 3 x})

Wende die Quotientenregel an:

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

= \frac{\frac{d}{d x} ( tan ( 20 x ) ) ( x ^{\frac{2}{3}} + 3 x ) - \frac{d}{d x} ( x ^{\frac{2}{3}} + 3 x ) tan ( 20 x )}{( x ^{\frac{2}{3}} + 3 x ) ^{2}}

\frac{d}{d x} (tan (20 x)) = sec^{2} (20 x) \cdot 20

\frac{d}{d x} (x^{\frac{2}{3}} + 3 x) = \frac{2}{3 x ^{\frac{1}{3}}} + 3

= \frac{sec ^{2} ( 20 x ) \cdot 20 ( x ^{\frac{2}{3}} + 3 x ) - ( \frac{2}{3 x ^{\frac{1}{3}}} + 3 ) tan ( 20 x )}{( x ^{\frac{2}{3}} + 3 x ) ^{2}}

Vereinfache

\frac{sec ^{2} ( 20 x ) \cdot 20 ( x ^{\frac{2}{3}} + 3 x ) - ( \frac{2}{3 x ^{\frac{1}{3}}} + 3 ) tan ( 20 x )}{( x ^{\frac{2}{3}} + 3 x ) ^{2}} : \frac{- 2 tan ( 20 x ) + 60 x sec ^{2} ( 20 x ) + 180 x ^{\frac{4}{3}} sec ^{2} ( 20 x ) - 9 x ^{\frac{1}{3}} tan ( 20 x )}{3 x ^{\frac{1}{3}} ( x ^{\frac{2}{3}} + 3 x ) ^{2}}

= \frac{- 2 tan ( 20 x ) + 60 x sec ^{2} ( 20 x ) + 180 x ^{\frac{4}{3}} sec ^{2} ( 20 x ) - 9 x ^{\frac{1}{3}} tan ( 20 x )}{3 x ^{\frac{1}{3}} ( x ^{\frac{2}{3}} + 3 x ) ^{2}}

x \to 0 lim (\frac{x}{ln ( x + 1 )})

\frac{\partial}{\partial x} (\frac{x ^{4} + y ^{4}}{x ^{2} + y ^{2}})

\frac{d}{d x} (5 sin^{3} (x))

f (x) = 2 x e^{x}

y^{^{'}} + x (y^{2} + y) = 0