Math Problem Statement
Solution
Let's solve each part of the question step-by-step.
Part (a): Find all second-order partial derivatives of
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First, calculate the first-order partial derivatives of :
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Partial derivative with respect to :
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Partial derivative with respect to :
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Now, calculate the second-order partial derivatives:
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Second partial derivative with respect to ( ):
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Second partial derivative with respect to ( ):
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Mixed partial derivative with respect to and ( ):
So, the second-order partial derivatives are:
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Part (b): Use the chain rule to evaluate , where , , and .
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Find the partial derivatives of with respect to and :
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With respect to :
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With respect to :
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Now, find the partial derivatives of and with respect to :
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For :
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For :
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Apply the chain rule to find : Substituting the values we found:
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Express and in terms of and if needed:
Thus, the final expression for is:
Would you like further clarification on any part of this solution?
Further Questions for Practice:
- How would you find third-order partial derivatives of ?
- Can you simplify the expression for further?
- What would change in part (b) if instead?
- How does the chain rule differ when there are three variables involved?
- How can you verify the mixed partial derivatives are equal, i.e., ?
Tip: Always double-check your substitution of terms when applying the chain rule to avoid common mistakes in calculations.
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Math Problem Analysis
Mathematical Concepts
Partial Derivatives
Chain Rule
Multivariable Calculus
Formulas
Second-order partial derivatives: \( f_{xx}, f_{yy}, f_{xy} \)
Chain rule for partial derivatives: \( \frac{\partial w}{\partial x} = \frac{\partial w}{\partial u} \cdot \frac{\partial u}{\partial x} + \frac{\partial w}{\partial s} \cdot \frac{\partial s}{\partial x} \)
Theorems
Mixed Partial Derivative Theorem (Schwarz's Theorem)
Suitable Grade Level
Undergraduate Calculus
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