Question

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

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

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)
$$\displaystyle{z}_{{{x}{y}}}={z}_{{{y}{x}}}$$