What is (\partial)/(\partial y)(-1/y sin(x/y)) ?

The answer to (\partial)/(\partial y)(-1/y sin(x/y)) is -(-xcos(x/y)-ysin(x/y))/(y^3)

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(\partial)/(\partial y)(-1/y sin(x/y))

\frac{\partial}{\partial y} (- \frac{1}{y} sin (\frac{x}{y}))

- \frac{- x cos ( \frac{x}{y} ) - y sin ( \frac{x}{y} )}{y ^{3}}

Solution steps

\frac{\partial}{\partial y} (- \frac{1}{y} sin (\frac{x}{y}))

Simplify

- \frac{1}{y} sin (\frac{x}{y}) : - \frac{sin ( \frac{x}{y} )}{y}

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

= - \frac{( - \frac{x c o s ( \frac{x}{y} )}{y ^{2}} ) y - 1 \cdot sin ( \frac{x}{y} )}{y ^{2}}

Simplify

- \frac{( - \frac{x c o s ( \frac{x}{y} )}{y ^{2}} ) y - 1 \cdot sin ( \frac{x}{y} )}{y ^{2}} : - \frac{- x cos ( \frac{x}{y} ) - y sin ( \frac{x}{y} )}{y ^{3}}

= - \frac{- x cos ( \frac{x}{y} ) - y sin ( \frac{x}{y} )}{y ^{3}}

\frac{d ^{2}}{d x ^{2}} (\frac{5}{6 x ^{4}})

\frac{d}{d x} ((x - 1) sin (x))

\frac{\partial}{\partial x} (ln (e^{x} + e^{y}))

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

\int x tan (π x) d x

What is (\partial)/(\partial y)(-1/y sin(x/y)) ?
The answer to (\partial)/(\partial y)(-1/y sin(x/y)) is -(-xcos(x/y)-ysin(x/y))/(y^3)