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

解

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

10 x^{9} sin^{5} (x) + 5 sin^{4} (x) cos (x) x^{10}

解答ステップ

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

簡素化

(x^{2} sin (x))^{5} : x^{10} sin^{5} (x)

= \frac{d}{d x} (x^{10} sin^{5} (x))

積の法則を適用:

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

f = x^{10}, g = sin^{5} (x)

= \frac{d}{d x} (x^{10}) sin^{5} (x) + \frac{d}{d x} (sin^{5} (x)) x^{10}

\frac{d}{d x} (x^{10}) = 10 x^{9}

\frac{d}{d x} (sin^{5} (x)) = 5 sin^{4} (x) cos (x)

= 10 x^{9} sin^{5} (x) + 5 sin^{4} (x) cos (x) x^{10}