derivative of (sin(4x)^{6x})

Lösung

\frac{d}{d x} ((sin (4 x))^{6 x})

6 sin^{6 x} (4 x) (ln (sin (4 x)) + 4 x cot (4 x))

Schritte zur Lösung

\frac{d}{d x} ((sin (4 x))^{6 x})

Wende Exponentenregel an:

a^{b} = e^{b l n (a)}

(sin (4 x))^{6 x} = e^{6 x l n (s i n (4 x))}

= \frac{d}{d x} (e^{6 x l n (s i n (4 x))})

Wende die Kettenregel an:

e^{6 x l n (s i n (4 x))} \frac{d}{d x} (6 x ln (sin (4 x)))

= e^{6 x l n (s i n (4 x))} \frac{d}{d x} (6 x ln (sin (4 x)))

\frac{d}{d x} (6 x ln (sin (4 x))) = 6 (ln (sin (4 x)) + 4 x cot (4 x))

= e^{6 x l n (s i n (4 x))} \cdot 6 (ln (sin (4 x)) + 4 x cot (4 x))

e^{6 x l n (s i n (4 x))} = sin^{6 x} (4 x)

= 6 sin^{6 x} (4 x) (ln (sin (4 x)) + 4 x cot (4 x))

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\int sin^{2} (k x) d x