Question

Find the four second partial derivatives. Observe that the second mixed partials are equal. z=e^{x}\tan y

Derivatives
ANSWERED
asked 2021-02-27
Find the four second partial derivatives. Observe that the second mixed partials are equal.
\(\displaystyle{z}={e}^{{{x}}}{\tan{{y}}}\)

Answers (1)

2021-03-01
Step 1
To find second partial derivatives of \(\displaystyle{z}={e}^{{{x}}}{\tan{{\left({y}\right)}}}\)
Step 2
Differentiate z with respect to x
\(\displaystyle{z}_{{{x}}}={e}^{{{x}}}{\tan{{\left({y}\right)}}}\)
Differentiate z with respect to y
\(\displaystyle{z}_{{{y}}}={e}^{{{x}}}{{\sec}^{{{2}}}{\left({y}\right)}}\)
Step 3
Differentiate \(\displaystyle{z}_{{{x}}}\) with respect to x
\(\displaystyle{z}_{{\times}}={e}^{{{x}}}{\tan{{\left({y}\right)}}}\)
Differentiate \(\displaystyle{z}_{{{y}}}\) with respect to x
\(\displaystyle{z}_{{{y}{x}}}={e}^{{{x}}}{{\sec}^{{{2}}}{\left({y}\right)}}\)...(1)
Step 4
Differentiate \(\displaystyle{z}_{{{x}}}\) with respect to y
\(\displaystyle{z}_{{{x}{y}}}={e}^{{{x}}}{{\sec}^{{{2}}}{\left({y}\right)}}\)...(2)
Differentiate z with respect to y
\(\displaystyle{z}_{{{y}{y}}}={e}^{{{x}}}{\left[{2}{\sec{{\left({y}\right)}}}\right]}{\left[{\sec{{\left({y}\right)}}}{\tan{{\left({y}\right)}}}\right]}\)
\(\displaystyle{z}_{{{y}{y}}}={2}{e}^{{{x}}}{{\sec}^{{{2}}}{\left({y}\right)}}{\tan{{\left({y}\right)}}}\)
From(1) and (2)
\(\displaystyle{z}_{{{x}{y}}}={z}_{{{y}{x}}}\)
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