derivative of (sqrt(y-x^2+2)/(x-y))

解答

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

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

求解步骤

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

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

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

\frac{d}{d x} (x - y) = 1 - \frac{d y}{d x}

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

化简

\frac{\frac{- 2 x + \frac{d y}{d x}}{2 y - x ^{2} + 2} ( x - y ) - ( 1 - \frac{d y}{d x} ) y - x ^{2} + 2}{( x - y ) ^{2}} : \frac{- 2 x ^{2} \frac{d y}{d x} + 2 x y + x \frac{d y}{d x} + y \frac{d y}{d x} - 2 y + 4 \frac{d y}{d x} - 4}{2 ( x - y ) ^{2} y - x ^{2} + 2}

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

y^{^{'}} = \frac{y ^{2} + 2}{x y - 4 y}

\frac{\partial}{\partial x} (x y e^{4 y})

t an g e n t y = x^{2} + 6 x - 5, at x = 2

\int x^{9} e^{x^{10}} d x

\frac{2 d y}{d x} + 7 y = 1