derivative 1/(\sqrt[3]{(4-x^3)^4)}

解答

导数

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

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

求解步骤

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

使用指数法则:

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

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

使用连锁律:

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

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

\frac{d}{d x} ((4 - x^{3})^{4}) = - 12 x^{2} (4 - x^{3})^{3}

= - \frac{1}{3 ( ( 4 - x ^{3} ) ^{4} ) ^{\frac{4}{3}}} (- 12 x^{2} (4 - x^{3})^{3})

化简

- \frac{1}{3 ( ( 4 - x ^{3} ) ^{4} ) ^{\frac{4}{3}}} (- 12 x^{2} (4 - x^{3})^{3}) : \frac{4 x ^{2} ( - x ^{3} + 4 ) ^{3}}{( ( 4 - x ^{3} ) ^{4} ) ^{\frac{4}{3}}}

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

\int \frac{3 x + 1}{x ^{2}} d x

\int_{1}^{e} ln (16 x) d x

x \to 1 lim (\frac{4 - 3 x}{1 - x})

\int_{1}^{1000} \frac{17}{x ^{3}} d x

\frac{d}{d t} (\frac{sec ^{2} ( t )}{cos ( t ) ( sec ( t ) )})