derivative of ln((5x^3sin(-5x)/(3-x)))

解答

\frac{d}{d x} (ln (\frac{5 x ^{3} sin ( - 5 x )}{3 - x}))

\frac{- 5 x ^{2} cos ( 5 x ) - 2 x sin ( 5 x ) + 15 x cos ( 5 x ) + 9 sin ( 5 x )}{x sin ( 5 x ) ( - x + 3 )}

求解步骤

\frac{d}{d x} (ln (\frac{5 x ^{3} sin ( - 5 x )}{3 - x}))

化简

ln (\frac{5 x ^{3} sin ( - 5 x )}{3 - x}) : ln (- \frac{5 x ^{3} sin ( 5 x )}{3 - x})

= \frac{d}{d x} (ln (- \frac{5 x ^{3} sin ( 5 x )}{3 - x}))

使用连锁律:

\frac{1}{- \frac{5 x ^{3} s i n ( 5 x )}{3 - x}} \frac{d}{d x} (- \frac{5 x ^{3} sin ( 5 x )}{3 - x})

= \frac{1}{- \frac{5 x ^{3} s i n ( 5 x )}{3 - x}} \frac{d}{d x} (- \frac{5 x ^{3} sin ( 5 x )}{3 - x})

\frac{d}{d x} (- \frac{5 x ^{3} sin ( 5 x )}{3 - x}) = - \frac{5 ( - 5 x ^{4} cos ( 5 x ) - 2 x ^{3} sin ( 5 x ) + 15 x ^{3} cos ( 5 x ) + 9 x ^{2} sin ( 5 x ) )}{( 3 - x ) ^{2}}

= \frac{1}{- \frac{5 x ^{3} s i n ( 5 x )}{3 - x}} (- \frac{5 ( - 5 x ^{4} cos ( 5 x ) - 2 x ^{3} sin ( 5 x ) + 15 x ^{3} cos ( 5 x ) + 9 x ^{2} sin ( 5 x ) )}{( 3 - x ) ^{2}})

化简

\frac{1}{- \frac{5 x ^{3} s i n ( 5 x )}{3 - x}} (- \frac{5 ( - 5 x ^{4} cos ( 5 x ) - 2 x ^{3} sin ( 5 x ) + 15 x ^{3} cos ( 5 x ) + 9 x ^{2} sin ( 5 x ) )}{( 3 - x ) ^{2}}) : \frac{- 5 x ^{2} cos ( 5 x ) - 2 x sin ( 5 x ) + 15 x cos ( 5 x ) + 9 sin ( 5 x )}{x sin ( 5 x ) ( - x + 3 )}

= \frac{- 5 x ^{2} cos ( 5 x ) - 2 x sin ( 5 x ) + 15 x cos ( 5 x ) + 9 sin ( 5 x )}{x sin ( 5 x ) ( - x + 3 )}

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