Math Problem Statement
Solution
To verify that for , let's proceed with finding the mixed partial derivatives.
Step-by-Step Solution
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Given Function: .
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First Partial Derivative with respect to :
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First Partial Derivative with respect to :
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Mixed Partial Derivative :
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Mixed Partial Derivative :
Since , the equality is verified.
Would you like further details on each differentiation step, or do you have any questions?
Here are 5 related questions that could expand your understanding:
- What are mixed partial derivatives, and when are they equal?
- How can the chain rule be applied in partial differentiation?
- How does one interpret the inverse trigonometric function in the context of partial derivatives?
- What are some physical applications of mixed partial derivatives?
- Can mixed partial derivatives be unequal, and under what conditions?
Tip: For inverse trigonometric functions, always check if simplification is possible by considering the function's argument structure before applying derivative rules directly.
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Math Problem Analysis
Mathematical Concepts
Calculus
Partial Derivatives
Inverse Trigonometric Functions
Mixed Partial Derivatives
Formulas
First partial derivative of \( \sin^{-1} \left( \frac{x}{y} \right) \) with respect to \( x \): \( \frac{1}{y \sqrt{y^2 - x^2}} \)
First partial derivative of \( \sin^{-1} \left( \frac{x}{y} \right) \) with respect to \( y \): \( -\frac{x}{y \sqrt{y^2 - x^2}} \)
Theorems
Clairaut's Theorem on Equality of Mixed Partials
Suitable Grade Level
Undergraduate Calculus
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