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

解答

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

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

求解步骤

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

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

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

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

化简

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

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

l a pl a ce t r an s f or m e^{- 3 t} sin (2 t)

\frac{d}{d x} (\frac{x ^{2}}{9})

y^{^{''}} + 4 y = 2 csc^{2} (2 t)

\frac{d}{d y} (2 y^{2} e^{x y^{2}})

a re a y = x^{2}, y = x