derivative of (9x/(64+x^2+y^2))

解答

\frac{d}{d x} (\frac{9 x}{64 + x ^{2} + y ^{2}})

\frac{9 ( - x ^{2} - 2 x y \frac{d y}{d x} + y ^{2} + 64 )}{( 64 + x ^{2} + y ^{2} ) ^{2}}

求解步骤

\frac{d}{d x} (\frac{9 x}{64 + x ^{2} + y ^{2}})

将常数提出:

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

= 9 \frac{d}{d x} (\frac{x}{64 + x ^{2} + y ^{2}})

使用微分除法定则:

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

= 9 \cdot \frac{\frac{d x}{d x} ( 64 + x ^{2} + y ^{2} ) - \frac{d}{d x} ( 64 + x ^{2} + y ^{2} ) x}{( 64 + x ^{2} + y ^{2} ) ^{2}}

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

\frac{d}{d x} (64 + x^{2} + y^{2}) = 2 x + 2 y \frac{d y}{d x}

= 9 \cdot \frac{1 \cdot ( 64 + x ^{2} + y ^{2} ) - ( 2 x + 2 y \frac{d y}{d x} ) x}{( 64 + x ^{2} + y ^{2} ) ^{2}}

化简

9 \cdot \frac{1 \cdot ( 64 + x ^{2} + y ^{2} ) - ( 2 x + 2 y \frac{d y}{d x} ) x}{( 64 + x ^{2} + y ^{2} ) ^{2}} : \frac{9 ( - x ^{2} - 2 x y \frac{d y}{d x} + y ^{2} + 64 )}{( 64 + x ^{2} + y ^{2} ) ^{2}}

= \frac{9 ( - x ^{2} - 2 x y \frac{d y}{d x} + y ^{2} + 64 )}{( 64 + x ^{2} + y ^{2} ) ^{2}}

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t \to 0 lim (cos (3 t))

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