derivative f(x)=((x^3)/(x^8+5))^2

解答

导数

f (x) = (\frac{x ^{3}}{x ^{8} + 5})^{2}

\frac{2 x ^{3} ( - 5 x ^{10} + 15 x ^{2} )}{( x ^{8} + 5 ) ^{3}}

求解步骤

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

使用连锁律:

2 \cdot \frac{x ^{3}}{x ^{8} + 5} \frac{d}{d x} (\frac{x ^{3}}{x ^{8} + 5})

= 2 \cdot \frac{x ^{3}}{x ^{8} + 5} \frac{d}{d x} (\frac{x ^{3}}{x ^{8} + 5})

\frac{d}{d x} (\frac{x ^{3}}{x ^{8} + 5}) = \frac{- 5 x ^{10} + 15 x ^{2}}{( x ^{8} + 5 ) ^{2}}

= 2 \cdot \frac{x ^{3}}{x ^{8} + 5} \cdot \frac{- 5 x ^{10} + 15 x ^{2}}{( x ^{8} + 5 ) ^{2}}

化简

2 \cdot \frac{x ^{3}}{x ^{8} + 5} \cdot \frac{- 5 x ^{10} + 15 x ^{2}}{( x ^{8} + 5 ) ^{2}} : \frac{2 x ^{3} ( - 5 x ^{10} + 15 x ^{2} )}{( x ^{8} + 5 ) ^{3}}

= \frac{2 x ^{3} ( - 5 x ^{10} + 15 x ^{2} )}{( x ^{8} + 5 ) ^{3}}