derivative f(x)=(x^2-2/x)^{-5}

解答

导数

f (x) = (x^{2} - \frac{2}{x})^{- 5}

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

求解步骤

\frac{d}{d x} ((x^{2} - \frac{2}{x})^{- 5})

使用连锁律:

- \frac{5}{( x ^{2} - \frac{2}{x} ) ^{6}} \frac{d}{d x} (x^{2} - \frac{2}{x})

= - \frac{5}{( x ^{2} - \frac{2}{x} ) ^{6}} \frac{d}{d x} (x^{2} - \frac{2}{x})

\frac{d}{d x} (x^{2} - \frac{2}{x}) = 2 x + \frac{2}{x ^{2}}

= - \frac{5}{( x ^{2} - \frac{2}{x} ) ^{6}} (2 x + \frac{2}{x ^{2}})

化简

- \frac{5}{( x ^{2} - \frac{2}{x} ) ^{6}} (2 x + \frac{2}{x ^{2}}) : - \frac{5 x ^{4} ( 2 x ^{3} + 2 )}{( x ^{3} - 2 ) ^{6}}

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

x \to \infty lim (13 x (e^{\frac{1}{x}}) - 13 x)

y^{^{'}} = 8 - \frac{y}{200} 4, y (0) = 50

x \to \frac{π}{4} lim (sin (2 x))

\int_{- 1}^{2} \frac{x ^{2}}{3} d x

\frac{d}{d x} (\frac{4 sin ( x )}{5 cos ( y )})