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

解答

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

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

求解步骤

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

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

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

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

\int (4 - x^{2}) d x

\frac{d}{d t} (cos (t) sin (t))

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

\int 7 x (8 - x) d x

a re a y^{2} = 4 + x, y^{2} + x = 2