What is (\partial}{\partial z}(ln(\frac{xy)/z)) ?

The answer to (\partial}{\partial z}(ln(\frac{xy)/z)) is -1/z

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(\partial}{\partial z}(ln(\frac{xy)/z))

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

- \frac{1}{z}

Solution steps

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

Treat

x, y

as constants

Apply the chain rule:

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

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

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

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

Simplify

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

= - \frac{1}{z}

y = x^{2}, x = \frac{y}{2}

t \to 0 lim (3 e^{2 t})

\int - 2 sin^{2} (x) d x

\frac{d y}{d x} + 2 x y = 4 x

What is (\partial}{\partial z}(ln(\frac{xy)/z)) ?
The answer to (\partial}{\partial z}(ln(\frac{xy)/z)) is -1/z