derivative ln((7x^2+8x+5)/(8x^3-1))

解答

导数

ln (\frac{7 x ^{2} + 8 x + 5}{8 x ^{3} - 1})

\frac{- 56 x ^{4} - 128 x ^{3} - 120 x ^{2} - 14 x - 8}{( 7 x ^{2} + 8 x + 5 ) ( 8 x ^{3} - 1 )}

求解步骤

\frac{d}{d x} (ln (\frac{7 x ^{2} + 8 x + 5}{8 x ^{3} - 1}))

使用连锁律:

\frac{1}{\frac{7 x ^{2} + 8 x + 5}{8 x ^{3} - 1}} \frac{d}{d x} (\frac{7 x ^{2} + 8 x + 5}{8 x ^{3} - 1})

= \frac{1}{\frac{7 x ^{2} + 8 x + 5}{8 x ^{3} - 1}} \frac{d}{d x} (\frac{7 x ^{2} + 8 x + 5}{8 x ^{3} - 1})

\frac{d}{d x} (\frac{7 x ^{2} + 8 x + 5}{8 x ^{3} - 1}) = \frac{- 56 x ^{4} - 128 x ^{3} - 120 x ^{2} - 14 x - 8}{( 8 x ^{3} - 1 ) ^{2}}

= \frac{1}{\frac{7 x ^{2} + 8 x + 5}{8 x ^{3} - 1}} \cdot \frac{- 56 x ^{4} - 128 x ^{3} - 120 x ^{2} - 14 x - 8}{( 8 x ^{3} - 1 ) ^{2}}

化简

\frac{1}{\frac{7 x ^{2} + 8 x + 5}{8 x ^{3} - 1}} \cdot \frac{- 56 x ^{4} - 128 x ^{3} - 120 x ^{2} - 14 x - 8}{( 8 x ^{3} - 1 ) ^{2}} : \frac{- 56 x ^{4} - 128 x ^{3} - 120 x ^{2} - 14 x - 8}{( 7 x ^{2} + 8 x + 5 ) ( 8 x ^{3} - 1 )}

= \frac{- 56 x ^{4} - 128 x ^{3} - 120 x ^{2} - 14 x - 8}{( 7 x ^{2} + 8 x + 5 ) ( 8 x ^{3} - 1 )}

d er i v a t i v e f (x) = x e^{x} at x

\int 8 x + 9 x d x

d er i v a t i v e f (x) = (t^{6} + 1)^{500}

t an g e n t - \frac{x}{( 25 - x ^{2} ) ^{\frac{1}{2}}}

d er i v a t i v e (125 + x)^{- \frac{1}{3}}