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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{d}{d x} (x e^{4 x} + cos (2 x))

\frac{d y}{d x} = y^{2} - 16

x \to 0 lim (f (x) - g (x))

x \to 3 - lim (e^{\frac{7}{x - 3}})

\int \frac{1}{x 6 x + 25} d x

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