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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

\int_{0}^{x} t^{2} + t^{4} d t

\frac{d y}{d x} = y^{2} - 4

\frac{\partial}{\partial x} (sin (5 π x + 2 y))

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

f (x) = x^{4} - 20 x^{2} + 64, at x = 2

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