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

解

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

\frac{4 sin ^{2} ( x ) + x sin ( 2 x )}{x sin ^{2} ( x )}

解答ステップ

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

連鎖法則を適用:

\frac{1}{x ^{4} sin ^{2} ( x )} \frac{d}{d x} (x^{4} sin^{2} (x))

= \frac{1}{x ^{4} sin ^{2} ( x )} \frac{d}{d x} (x^{4} sin^{2} (x))

\frac{d}{d x} (x^{4} sin^{2} (x)) = 4 x^{3} sin^{2} (x) + sin (2 x) x^{4}

= \frac{1}{x ^{4} sin ^{2} ( x )} (4 x^{3} sin^{2} (x) + sin (2 x) x^{4})

簡素化

\frac{1}{x ^{4} sin ^{2} ( x )} (4 x^{3} sin^{2} (x) + sin (2 x) x^{4}) : \frac{4 sin ^{2} ( x ) + x sin ( 2 x )}{x sin ^{2} ( x )}

= \frac{4 sin ^{2} ( x ) + x sin ( 2 x )}{x sin ^{2} ( x )}