d/(dy)(xysin(y)+xy^2sin(x)+1/x)

解答

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

x (y cos (y) + sin (y)) + 2 x y sin (x)

求解步骤

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

将

x

视为常数

使用微分加减法定则:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

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

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

\frac{d}{d y} (\frac{1}{x}) = 0

= x (sin (y) + y cos (y)) + 2 x y sin (x) + 0

化简

= x (y cos (y) + sin (y)) + 2 x y sin (x)

x \to 3 lim (csc (\frac{π x}{4}))

\int e^{- x} \cdot cos (x) d x

(x y^{3} + 1) d x + x^{2} y^{2} d y = 0

x \frac{d y}{d x} - x^{2} y = - x^{2}

\frac{y ^{^{'}} - e ^{- t} + 4}{y} = - 4, y (0) = 3