(\partial)/(\partial x)(sqrt(xy+y/x))

解答

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

\frac{x ^{2} y - y}{2 x ^{\frac{3}{2}} x ^{2} y + y}

求解步骤

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

将

y

视为常数

使用连锁律:

\frac{1}{2 x y + \frac{y}{x}} \frac{\partial}{\partial x} (x y + \frac{y}{x})

= \frac{1}{2 x y + \frac{y}{x}} \frac{\partial}{\partial x} (x y + \frac{y}{x})

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

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

化简

\frac{1}{2 x y + \frac{y}{x}} (y - \frac{y}{x ^{2}}) : \frac{x ^{2} y - y}{2 x ^{\frac{3}{2}} x ^{2} y + y}

= \frac{x ^{2} y - y}{2 x ^{\frac{3}{2}} x ^{2} y + y}

\int \frac{cos ^{5} ( 5 x + 1 )}{sin ^{4} ( 5 x + 1 )} d x

t an g e n t y = \frac{( x - 1 )}{x + 1}

d er i v a t i v e - \frac{5}{7 x + 5}

x \to 0 + lim ((1 + \frac{1}{x})^{x})

\int sin (π t) d t