derivative of (3+x{f}(x)/(sqrt(x)))

解答

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

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

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

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

化简

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

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

d er i v a t i v e (arctan (4 x))^{2}

2 \frac{d ^{2} y}{d x ^{2}} + 8 y = cos (2 x)

\frac{\partial}{\partial x} (2 (x - 1))

\int \frac{1}{( x + 3 ) ^{3}} d x

\int_{- 1}^{2} (y + 2) - (y^{2}) d y