derivative of 2sec(2xtan(2x))

Lösung

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

2 (2 tan^{2} (2 x) sec (2 x) + 2 sec^{3} (2 x))

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die Produktregel an:

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

f = sec (2 x), g = tan (2 x)

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

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

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

= 2 (sec (2 x) tan (2 x) \cdot 2 tan (2 x) + sec^{2} (2 x) \cdot 2 sec (2 x))

Vereinfache

2 (sec (2 x) tan (2 x) \cdot 2 tan (2 x) + sec^{2} (2 x) \cdot 2 sec (2 x)) : 2 (2 tan^{2} (2 x) sec (2 x) + 2 sec^{3} (2 x))

= 2 (2 tan^{2} (2 x) sec (2 x) + 2 sec^{3} (2 x))

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