derivative of (6x+y/(x^2+y^2))

解答

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

\frac{- 6 x ^{2} + x ^{2} \frac{d y}{d x} - 2 x y - 12 x y \frac{d y}{d x} + 6 y ^{2} - y ^{2} \frac{d y}{d x}}{( x ^{2} + y ^{2} ) ^{2}}

求解步骤

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

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} ( 6 x + y ) ( x ^{2} + y ^{2} ) - \frac{d}{d x} ( x ^{2} + y ^{2} ) ( 6 x + y )}{( x ^{2} + y ^{2} ) ^{2}}

\frac{d}{d x} (6 x + y) = 6 + \frac{d y}{d x}

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

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

乘开

(6 + \frac{d y}{d x}) (x^{2} + y^{2}) - (2 x + 2 y \frac{d y}{d x}) (6 x + y) : - 6 x^{2} + x^{2} \frac{d y}{d x} - 2 x y - 12 x y \frac{d y}{d x} + 6 y^{2} - y^{2} \frac{d y}{d x}

= \frac{- 6 x ^{2} + x ^{2} \frac{d y}{d x} - 2 x y - 12 x y \frac{d y}{d x} + 6 y ^{2} - y ^{2} \frac{d y}{d x}}{( x ^{2} + y ^{2} ) ^{2}}

x \to 3 lim (\frac{x - 3}{x ^{2} - 3})

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

f (x) = e^{x} - e^{- x}

l a pl a ce t r an s f or m f (t) = (t + 1)^{3}

s l o p e (- 1, - 1), (1, - 4)