If ƒ(x, y) = x¹⁰ sin(cos(yesin(cos(arctan(ry)))) + xy³, then what is fry (x, y) – fyx fxy(x, (x, y)? 0 ○ fxx (x, y) XX fyy (x, y) fyry (x, y) fryx (x, y)
If ƒ(x, y) = x¹⁰ sin(cos(yesin(cos(arctan(ry)))) + xy³, then what is fry (x, y) – fyx fxy(x, (x, y)? 0 ○ fxx (x, y) XX fyy (x, y) fyry (x, y) fryx (x, y)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![The problem is exploring higher-order partial derivatives. It presents a function and asks for the expression \( f_{xy}(x, y) - f_{yx}(x, y) \).
### Problem Statement:
If
\[ f(x, y) = x^{10} \sin \left( \cos \left( y^{\sin(\cos(\arctan(xy)))} \right) \right) + xy^3, \]
then what is \( f_{xy}(x, y) - f_{yx}(x, y) \)?
### Options:
- \( \circ \) \( 0 \)
- \( \circ \) \( f_{xx}(x, y) \)
- \( \circ \) \( f_{yy}(x, y) \)
- \( \circ \) \( f_{yxy}(x, y) \)
- \( \circ \) \( f_{xyx}(x, y) \)
### Explanation:
The expression \( f_{xy}(x, y) - f_{yx}(x, y) \) is dependent on the mixed partial derivatives of the function \( f(x, y) \). Typically, if \( f \) is sufficiently smooth (i.e., continuous partial derivatives), Clairaut's theorem assures that \( f_{xy} = f_{yx} \), making their difference zero. Hence, the correct choice is likely:
- \( \circ \) \( 0 \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F72514fec-ab1d-4e58-a749-feb22f2ce6e9%2F2ddd3bdf-0fe6-43e8-a771-134df8ddaac9%2Flz7nyjw_processed.png&w=3840&q=75)
Transcribed Image Text:The problem is exploring higher-order partial derivatives. It presents a function and asks for the expression \( f_{xy}(x, y) - f_{yx}(x, y) \).
### Problem Statement:
If
\[ f(x, y) = x^{10} \sin \left( \cos \left( y^{\sin(\cos(\arctan(xy)))} \right) \right) + xy^3, \]
then what is \( f_{xy}(x, y) - f_{yx}(x, y) \)?
### Options:
- \( \circ \) \( 0 \)
- \( \circ \) \( f_{xx}(x, y) \)
- \( \circ \) \( f_{yy}(x, y) \)
- \( \circ \) \( f_{yxy}(x, y) \)
- \( \circ \) \( f_{xyx}(x, y) \)
### Explanation:
The expression \( f_{xy}(x, y) - f_{yx}(x, y) \) is dependent on the mixed partial derivatives of the function \( f(x, y) \). Typically, if \( f \) is sufficiently smooth (i.e., continuous partial derivatives), Clairaut's theorem assures that \( f_{xy} = f_{yx} \), making their difference zero. Hence, the correct choice is likely:
- \( \circ \) \( 0 \)
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