derivative of (sin(x)^{3x})

Lösung

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

3 sin^{3 x} (x) (ln (sin (x)) + x cot (x))

Schritte zur Lösung

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

Wende Exponentenregel an:

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

(sin (x))^{3 x} = e^{3 x l n (s i n (x))}

= \frac{d}{d x} (e^{3 x l n (s i n (x))})

Wende die Kettenregel an:

e^{3 x l n (s i n (x))} \frac{d}{d x} (3 x ln (sin (x)))

= e^{3 x l n (s i n (x))} \frac{d}{d x} (3 x ln (sin (x)))

\frac{d}{d x} (3 x ln (sin (x))) = 3 (ln (sin (x)) + x cot (x))

= e^{3 x l n (s i n (x))} \cdot 3 (ln (sin (x)) + x cot (x))

e^{3 x l n (s i n (x))} = sin^{3 x} (x)

= 3 sin^{3 x} (x) (ln (sin (x)) + x cot (x))

\int sin^{5} (4 x) d x

x \to 9 lim (\frac{1}{x ^{2}})

\int x^{2} ln (4 x) d x

x \to - \frac{π}{4} lim (sec (x))

x \to 1 lim (+ (\frac{1}{x ^{2} - 1}))