derivative of (1-2x/(3x^2))

解答

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

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

求解步骤

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

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

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

使用微分除法定则:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{1}{3} \cdot \frac{\frac{d}{d x} ( 1 - 2 x ) x ^{2} - \frac{d}{d x} ( x ^{2} ) ( 1 - 2 x )}{( x ^{2} ) ^{2}}

\frac{d}{d x} (1 - 2 x) = - 2

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

= \frac{1}{3} \cdot \frac{( - 2 ) x ^{2} - 2 x ( 1 - 2 x )}{( x ^{2} ) ^{2}}

化简

\frac{1}{3} \cdot \frac{( - 2 ) x ^{2} - 2 x ( 1 - 2 x )}{( x ^{2} ) ^{2}} : - \frac{2 ( - x + 1 )}{3 x ^{3}}

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

x \to 1 lim (\frac{2}{x ^{3} - x})

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

\frac{d ^{2}}{d x ^{2}} (e^{\frac{6}{x}})

x \to \infty lim (- 3 x^{3} + x)

y = 0, y = 3 (x^{3} - x)