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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Simplify

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

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

\int \frac{y}{( 4 + y ^{2} )} d y

\frac{d}{d x} (lo g_{3} ((x^{2} + 2 x)^{4}))

\int x 2^{x + 2} d x

\frac{d ^{2} y}{d x ^{2}} - 9 \frac{d y}{d x} + 4 y = x e^{x}

sin (x) + \frac{3}{5} cot (x)

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