partial derivative of ln(x*y/z) for z

1) Differentiate with respect to z while holding x, y constant

Let's perform the differentiation:

\frac{d}{d z} (\ln (x \cdot \frac{y}{z}))

We differentiate using:

\frac{d}{d z} (\ln (f)) = \frac{d}{d z} (f) \cdot \frac{1}{f}

(where f =

\frac{x y}{z}

)

\frac{d}{d z} (\frac{x y}{z}) \cdot \frac{1}{\frac{x y}{z}}

For

u1

x y

and

v2

z

, use the product/quotient rule:

\frac{d}{d z} (\frac{u1}{v2}) = \frac{\frac{d}{d z} (u1) \cdot v2 - u1 \cdot \frac{d}{d z} (v2)}{{v2}^{2}}

2) Let's perform the differentiation

u1

\frac{d}{d z} (x y)

0

3) Let's differentiate

v2

\frac{d}{d z} (z)

1

4) Let's evaluate the resulting derivative (

\frac{u1}{v2}

)':

\frac{0 \cdot z - x y \cdot 1}{z^{2}}

\frac{0 - x y}{z^{2}}

- \frac{x y}{z^{2}}

5) Then you can continue the differentiation:

- \frac{\frac{x y}{z^{2}}}{\frac{x y}{z}}

- \frac{x y}{z^{2}} \cdot \frac{z}{x y}

- \frac{x y z}{z^{2} x y}

We simplify the fraction:

\frac{z}{z^{2}}

by the common factor:

z

to get the new quotient:

\frac{1}{z}

▶Derivative:

- \frac{1}{z}