derivative ((4x^3+5)/(-5x^2+1))^{1/4}

解答

导数

(\frac{4 x ^{3} + 5}{- 5 x ^{2} + 1})^{\frac{1}{4}}

\frac{x ( - 10 x ^{3} + 6 x + 25 )}{2 ( 4 x ^{3} + 5 ) ^{\frac{3}{4}} ( - 5 x ^{2} + 1 ) ^{\frac{5}{4}}}

求解步骤

\frac{d}{d x} ((\frac{4 x ^{3} + 5}{- 5 x ^{2} + 1})^{\frac{1}{4}})

使用连锁律:

\frac{1}{4 ( \frac{4 x ^{3} + 5}{- 5 x ^{2} + 1} ) ^{\frac{3}{4}}} \frac{d}{d x} (\frac{4 x ^{3} + 5}{- 5 x ^{2} + 1})

= \frac{1}{4 ( \frac{4 x ^{3} + 5}{- 5 x ^{2} + 1} ) ^{\frac{3}{4}}} \frac{d}{d x} (\frac{4 x ^{3} + 5}{- 5 x ^{2} + 1})

\frac{d}{d x} (\frac{4 x ^{3} + 5}{- 5 x ^{2} + 1}) = \frac{- 20 x ^{4} + 12 x ^{2} + 50 x}{( - 5 x ^{2} + 1 ) ^{2}}

= \frac{1}{4 ( \frac{4 x ^{3} + 5}{- 5 x ^{2} + 1} ) ^{\frac{3}{4}}} \cdot \frac{- 20 x ^{4} + 12 x ^{2} + 50 x}{( - 5 x ^{2} + 1 ) ^{2}}

化简

\frac{1}{4 ( \frac{4 x ^{3} + 5}{- 5 x ^{2} + 1} ) ^{\frac{3}{4}}} \cdot \frac{- 20 x ^{4} + 12 x ^{2} + 50 x}{( - 5 x ^{2} + 1 ) ^{2}} : \frac{x ( - 10 x ^{3} + 6 x + 25 )}{2 ( 4 x ^{3} + 5 ) ^{\frac{3}{4}} ( - 5 x ^{2} + 1 ) ^{\frac{5}{4}}}

= \frac{x ( - 10 x ^{3} + 6 x + 25 )}{2 ( 4 x ^{3} + 5 ) ^{\frac{3}{4}} ( - 5 x ^{2} + 1 ) ^{\frac{5}{4}}}