Question

# Find the four second partial derivatives. Observe that the second mixed partials are equal. z=\ln(x-y)

Derivatives
Find the four second partial derivatives. Observe that the second mixed partials are equal.
$$\displaystyle{z}={\ln{{\left({x}-{y}\right)}}}$$

2021-05-04
Step 1
$$\displaystyle{z}={\ln{{\left({x}-{y}\right)}}}$$
$$\displaystyle{\frac{{{\left.{d}{z}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{{1}}}{{{x}-{y}}}}$$
$$\displaystyle{\frac{{{d}^{{{2}}}{z}}}{{{\left.{d}{x}\right.}^{{{2}}}}}}={\frac{{-{1}}}{{{\left({x}-{y}\right)}^{{{2}}}}}}$$...(1)
Step 2
$$\displaystyle{\frac{{{\left.{d}{z}\right.}}}{{{\left.{d}{y}\right.}}}}={\frac{{-{1}}}{{{x}-{y}}}}$$
$$\displaystyle{\frac{{{d}^{{{2}}}{z}}}{{{\left.{d}{y}\right.}}}}={\frac{{-{1}}}{{{x}-{y}}}}$$
$$\displaystyle{\frac{{{d}^{{{2}}}{z}}}{{{\left.{d}{y}\right.}^{{{2}}}}}}={\frac{{-{1}}}{{{\left({x}-{y}\right)}^{{{2}}}}}}\ldots{\left({2}\right)}$$
From eq(1) and (2)
$$\displaystyle{\frac{{{d}^{{{2}}}{z}}}{{{\left.{d}{x}\right.}^{{{2}}}}}}={\frac{{{d}^{{{2}}}{z}}}{{{\left.{d}{y}\right.}^{{{2}}}}}}$$