What is (\partial)/(\partial z)((xy)/(x+y+z)) ?

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

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(\partial)/(\partial z)((xy)/(x+y+z))

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

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

Solution steps

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

Treat

x, y

as constants

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{d}{d x} (\frac{cos ( x )}{4 x})

x \to 1 - lim (x^{3} - 4)

\int 5 - 3 (1 + x^{2})^{- 1} d x

y = (arctan (8 x))^{2}

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

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