derivative of-(4x/((x^2-4)^2))

解答

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

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

求解步骤

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

将常数提出:

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

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

使用微分除法定则:

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

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

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

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

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

化简

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

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