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

The answer to (\partial)/(\partial x)(sin(y/x)) is -(ycos(y/x))/(x^2)

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

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

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

Solution steps

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

Treat

y

as a constant

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}}

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

Simplify

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

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

x \to - 2 lim (2 x^{3} - x^{2} + 7 x - 3)

(2.6) (- 2.6)

x \to 2 lim (\frac{\frac{2}{x} - 1}{2 - x})

\int \frac{x ^{2} - 2 x - 1}{( x - 1 ) ^{2} ( x ^{2} + 1 )} d x

\int \frac{( e ^{2 x} )}{e ^{x} + 1} d x

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