derivative of (sin(5x)(arccsc(2)x^3))

解

\frac{d}{d x} ((sin (5 x)) (arccsc (2) x^{3}))

\frac{π}{6} (5 x^{3} cos (5 x) + 3 x^{2} sin (5 x))

解答ステップ

\frac{d}{d x} ((sin (5 x)) (arccsc (2) x^{3}))

定数を除去:

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

= arccsc (2) \frac{d}{d x} (sin (5 x) x^{3})

積の法則を適用:

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

f = sin (5 x), g = x^{3}

= arccsc (2) (\frac{d}{d x} (sin (5 x)) x^{3} + \frac{d}{d x} (x^{3}) sin (5 x))

\frac{d}{d x} (sin (5 x)) = cos (5 x) \cdot 5

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

= arccsc (2) (cos (5 x) \cdot 5 x^{3} + 3 x^{2} sin (5 x))

次の自明恒等式を使用する:

arccsc (2) = \frac{π}{6}

x 22 \frac{2 3}{3} 1 arccsc (x) \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arccsc (x) 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{6} (5 x^{3} cos (5 x) + 3 x^{2} sin (5 x))