What is (\partial)/(\partial x)(y/((x+y)^2)) ?

The answer to (\partial)/(\partial x)(y/((x+y)^2)) is -(2y)/((x+y)^3)

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(\partial)/(\partial x)(y/((x+y)^2))

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

- \frac{2}{( x + y ) ^{3}} \frac{\partial}{\partial x} (x + y)

= - \frac{2}{( x + y ) ^{3}} \frac{\partial}{\partial x} (x + y)

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

= y (- \frac{2}{( x + y ) ^{3}} \cdot 1)

Simplify

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

= - \frac{2 y}{( x + y ) ^{3}}

\frac{d}{d x} (6 x^{2} - x^{3})

7 x^{2} - x^{3}

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

x \to 0 lim ((1 - x)^{\frac{1}{x}})

\frac{d}{d t} (- s t)

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