derivative f(x)=(8x)/(8+sqrt(x))

解答

导数

f (x) = \frac{8 x}{8 + x}

\frac{4 ( x + 16 )}{( 8 + x ) ^{2}}

求解步骤

\frac{d}{d x} (\frac{8 x}{8 + x})

将常数提出:

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

= 8 \frac{d}{d x} (\frac{x}{8 + x})

使用微分除法定则:

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

= 8 \cdot \frac{\frac{d x}{d x} ( 8 + x ) - \frac{d}{d x} ( 8 + x ) x}{( 8 + x ) ^{2}}

\frac{d x}{d x} = 1

\frac{d}{d x} (8 + x) = \frac{1}{2 x}

= 8 \cdot \frac{1 \cdot ( 8 + x ) - \frac{1}{2 x} x}{( 8 + x ) ^{2}}

化简

8 \cdot \frac{1 \cdot ( 8 + x ) - \frac{1}{2 x} x}{( 8 + x ) ^{2}} : \frac{4 ( x + 16 )}{( 8 + x ) ^{2}}

= \frac{4 ( x + 16 )}{( 8 + x ) ^{2}}

y^{^{'}} = \frac{( 1 + y ^{6} )}{x y ^{5}}

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