What is (\partial)/(\partial z)((x+y)/(z-3y)) ?

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

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(\partial)/(\partial z)((x+y)/(z-3y))

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

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

Solution steps

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

Treat

x, y

as constants

Take the constant out:

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

= (x + y) \frac{\partial}{\partial z} (\frac{1}{z - 3 y})

Apply exponent rule:

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

= (x + y) \frac{\partial}{\partial z} ((z - 3 y)^{- 1})

Apply the chain rule:

- \frac{1}{( z - 3 y ) ^{2}} \frac{\partial}{\partial z} (z - 3 y)

= - \frac{1}{( z - 3 y ) ^{2}} \frac{\partial}{\partial z} (z - 3 y)

\frac{\partial}{\partial z} (z - 3 y) = 1

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

Simplify

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

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

\frac{\partial}{\partial x} (3 \cdot (y + 4)^{3} \cdot (x - 2))

\int_{1}^{5} x d x

\frac{d}{d x} (2 4 - x^{2})

\frac{\partial}{\partial x} (\frac{4 - z ^{2}}{3 x})

\int \frac{x ^{2} + 3 x - 5}{x - 3} d x

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