Question

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

Derivatives
ANSWERED
asked 2021-05-03
Find the four second partial derivatives. Observe that the second mixed partials are equal.
\(\displaystyle{z}={\ln{{\left({x}-{y}\right)}}}\)

Answers (1)

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)
\(\displaystyle{\frac{{{d}^{{{2}}}{z}}}{{{\left.{d}{x}\right.}^{{{2}}}}}}={\frac{{{d}^{{{2}}}{z}}}{{{\left.{d}{y}\right.}^{{{2}}}}}}\)
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