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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{d}{d x} (\frac{cot ( x )}{x ^{2}})

\frac{d y}{d x} + \frac{1}{x} y = 7 x^{2}

\frac{d}{d x} (\frac{6}{x ^{2} + 1})

\frac{d}{d t} (1 - 3 t)

x e^{- x^{2}}

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