derivative of (cos(3x^3)/(tan(3x^2+2x)))

Lösung

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

- 9 x^{2} sin (3 x^{3}) cot (3 x^{2} + 2 x) - csc^{2} (3 x^{2} + 2 x) cos (3 x^{3}) (6 x + 2)

Schritte zur Lösung

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

Vereinfache

\frac{cos ( 3 x ^{3} )}{tan ( 3 x ^{2} + 2 x )} : cos (3 x^{3}) cot (3 x^{2} + 2 x)

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

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = cos (3 x^{3}), g = cot (3 x^{2} + 2 x)

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

\frac{d}{d x} (cos (3 x^{3})) = - sin (3 x^{3}) \cdot 9 x^{2}

\frac{d}{d x} (cot (3 x^{2} + 2 x)) = - csc^{2} (3 x^{2} + 2 x) (6 x + 2)

= (- sin (3 x^{3}) \cdot 9 x^{2}) cot (3 x^{2} + 2 x) + (- csc^{2} (3 x^{2} + 2 x) (6 x + 2)) cos (3 x^{3})

Vereinfache

= - 9 x^{2} sin (3 x^{3}) cot (3 x^{2} + 2 x) - csc^{2} (3 x^{2} + 2 x) cos (3 x^{3}) (6 x + 2)

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