Best solutions to - "Find the four second partial derivatives. Observe that..."

Kyran Hudson

Answered question

2021-02-27

Find the four second partial derivatives. Observe that the second mixed partials are equal.

z = e^{x} \tan y

Answer & Explanation

Latisha Oneil

Skilled2021-03-01Added 100 answers

Step 1
To find second partial derivatives of $z = e^{x} \tan (y)$
Step 2
Differentiate z with respect to x
$z_{x} = e^{x} \tan (y)$
Differentiate z with respect to y
$z_{y} = e^{x} \sec^{2} (y)$
Step 3
Differentiate $z_{x}$ with respect to x
$z_{x x} = e^{x} \tan (y)$
Differentiate $z_{y}$ with respect to x
$z_{y x} = e^{x} \sec^{2} (y)$ ...(1)
Step 4
Differentiate $z_{x}$ with respect to y
$z_{x y} = e^{x} \sec^{2} (y)$ ...(2)
Differentiate z with respect to y
$z_{y y} = e^{x} [2 \sec (y)] [\sec (y) \tan (y)]$
$z_{y y} = 2 e^{x} \sec^{2} (y) \tan (y)$
From(1) and (2)
$z_{x y} = z_{y x}$

Jeffrey Jordon

Expert2022-01-31Added 2605 answers

Answer is given below (on video)

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