derivative of 8/(sqrt((9x^2-7^5)))

解答

\frac{d}{d x} \frac{8}{( 9 x ^{2} - 7 ) ^{5}}

- \frac{360 x}{( 9 x ^{2} - 7 ) ^{\frac{7}{2}}}

求解步骤

\frac{d}{d x} \frac{8}{( 9 x ^{2} - 7 ) ^{5}}

将常数提出:

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

= 8 \frac{d}{d x} \frac{1}{( 9 x ^{2} - 7 ) ^{5}}

使用指数法则:

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

= 8 \frac{d}{d x} (((9 x^{2} - 7)^{5})^{- \frac{1}{2}})

使用连锁律:

- \frac{1}{2 ( ( 9 x ^{2} - 7 ) ^{5} ) ^{\frac{3}{2}}} \frac{d}{d x} ((9 x^{2} - 7)^{5})

= - \frac{1}{2 ( ( 9 x ^{2} - 7 ) ^{5} ) ^{\frac{3}{2}}} \frac{d}{d x} ((9 x^{2} - 7)^{5})

\frac{d}{d x} ((9 x^{2} - 7)^{5}) = 90 x (9 x^{2} - 7)^{4}

= 8 - \frac{1}{2 ( ( 9 x ^{2} - 7 ) ^{5} ) ^{\frac{3}{2}}} \cdot 90 x (9 x^{2} - 7)^{4}

化简

8 - \frac{1}{2 ( ( 9 x ^{2} - 7 ) ^{5} ) ^{\frac{3}{2}}} \cdot 90 x (9 x^{2} - 7)^{4} : - \frac{360 x}{( 9 x ^{2} - 7 ) ^{\frac{7}{2}}}

= - \frac{360 x}{( 9 x ^{2} - 7 ) ^{\frac{7}{2}}}

(x^{2} + 1) \frac{d y}{d x} + 3 x y = 6 x

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