derivative y=(3+xf(x))/(sqrt(x))

解答

导数

y = \frac{3 + x f ( x )}{x}

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

求解步骤

\frac{d}{d x} (\frac{3 + x f ( x )}{x})

使用微分除法定则:

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

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

\frac{d}{d x} (3 + x f (x)) = x \frac{df}{d x} + f (x)

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

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

化简

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

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

x \to - \infty lim (- 1)

\int_{- 3}^{4} ∣ x ∣ d x

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