Resolved: "Compute all of the second-order partial derivatives for..."

Marvin Mccormick

Answered question

2021-05-12

Compute all of the second-order partial derivatives for the functions and show that the mixed partial derivatives are equal.

f (x, y) = e^{x} \sin (x y)

Bertha Stark

Skilled2021-05-14Added 96 answers

Step 1
Given function is:

f (x, y) = e^{x} \sin (x y)

Differentiating f(x, y) partially with respect to x we get,

f_{x} (x, y) = e^{x} \sin (x y) + y e^{x} \cos (x y) = e^{x} [\sin (x y) + y \cos (x y)]

Differentiating f(x, y) partially with respect to y we get,

f_{y} (x, y) = x e^{x} \cos (x y)

Step 2
Moreover we get,

f_{x y} (x, y) = x e^{x} \cos (x y) + e^{x} \cos (x y) - x y e^{x} \sin (x y)

f_{y x} (x, y) = e^{x} \cos (x y) + x e^{x} \cos (x y) - x y e^{x} \sin (x y)

These are the second order partial derivatives of f(x, y).

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