derivative of 2/(3x(\sqrt[3]{x^2+1)})

解答

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

- \frac{2 ( 3 3 x ^{2} + 3 + 2 x ^{\frac{2}{3}} )}{9 x ^{2} ( 3 x ^{2} + 1 ) ^{2}}

求解步骤

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

将常数提出:

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

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

使用指数法则:

\frac{1}{a} = a^{- 1}

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

使用连锁律:

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

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

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

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

化简

\frac{2}{3} - \frac{1}{( x ( 3 x ^{2} + 1 ) ) ^{2}} (3 x^{2} + 1 + \frac{2 x ^{2}}{3 ( x ^{2} ) ^{\frac{2}{3}}}) : - \frac{2 ( 3 3 x ^{2} + 3 + 2 x ^{\frac{2}{3}} )}{9 x ^{2} ( 3 x ^{2} + 1 ) ^{2}}

= - \frac{2 ( 3 3 x ^{2} + 3 + 2 x ^{\frac{2}{3}} )}{9 x ^{2} ( 3 x ^{2} + 1 ) ^{2}}

\frac{( x ^{2} + 1 ) d y}{d x} + 8 x (y - 1) = 0

\int \frac{x + 4}{x} d x

d er i v a t i v e f (x) = tan (\frac{π}{3})

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

\frac{\partial}{\partial y} (\frac{x}{9 + y})