What is (\partial)/(\partial x)(sqrt(xy+y/x)) ?

The answer to (\partial)/(\partial x)(sqrt(xy+y/x)) is (x^2y-y)/(2x^{3/2)sqrt(x^2y+y)}

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

\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}})

Simplify

\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

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

- \frac{5}{7 x + 5}

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

\int sin (π t) d t

What is (\partial)/(\partial x)(sqrt(xy+y/x)) ?
The answer to (\partial)/(\partial x)(sqrt(xy+y/x)) is (x^2y-y)/(2x^{3/2)sqrt(x^2y+y)}