derivative of 4/(sqrt(1+x))

解答

\frac{d}{d x} (\frac{4}{1 + x})

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

求解步骤

\frac{d}{d x} (\frac{4}{1 + x})

将常数提出:

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

= 4 \frac{d}{d x} (\frac{1}{1 + x})

使用指数法则:

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

= 4 \frac{d}{d x} ((1 + x)^{- \frac{1}{2}})

使用连锁律:

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

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

\frac{d}{d x} (1 + x) = 1

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

化简

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

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

\int \frac{8}{x ( ln ( x ) + 1 ) ^{\frac{3}{2}}} d x

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