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

The answer to (\partial)/(\partial x)(ysin(y/x)) is -(y^2cos(y/x))/(x^2)

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(\partial)/(\partial x)(ysin(y/x))

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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What is (\partial)/(\partial x)(ysin(y/x)) ?
The answer to (\partial)/(\partial x)(ysin(y/x)) is -(y^2cos(y/x))/(x^2)