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

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

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

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

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

Solution steps

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

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})

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{d}{d x} (\frac{sin ( x ) cos ( x )}{2 x ^{2}})

x \to \infty + lim (5 + \frac{6}{x})

2 \frac{d ^{2} y}{d x ^{2}} - 14 \frac{d y}{d x} + 24 y = 3 sin (4 x)

x \to 2 - lim (e^{\frac{3}{( 2 - x )}})

\frac{d y}{d x} = 6 y e^{x + 2}

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