derivative of (x^2ln(x)^4)

解

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

8 x^{7} ln^{4} (x) + 4 x^{7} ln^{3} (x)

解答ステップ

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

簡素化

(x^{2} ln (x))^{4} : x^{8} ln^{4} (x)

= \frac{d}{d x} (x^{8} ln^{4} (x))

積の法則を適用:

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

f = x^{8}, g = ln^{4} (x)

= \frac{d}{d x} (x^{8}) ln^{4} (x) + \frac{d}{d x} (ln^{4} (x)) x^{8}

\frac{d}{d x} (x^{8}) = 8 x^{7}

\frac{d}{d x} (ln^{4} (x)) = \frac{4 ln ^{3} ( x )}{x}

= 8 x^{7} ln^{4} (x) + \frac{4 ln ^{3} ( x )}{x} x^{8}

\frac{4 ln ^{3} ( x )}{x} x^{8} = 4 x^{7} ln^{3} (x)

= 8 x^{7} ln^{4} (x) + 4 x^{7} ln^{3} (x)