Math Problem Statement
Differentiate implicitly to find
StartFraction d squared y Over dx squared EndFractiond2ydx2. 2y^2-xy+x^2=8
Solution
To differentiate implicitly and find , follow these steps:
Step 1: Differentiate the equation implicitly
The given equation is:
Differentiating both sides with respect to , keeping in mind that is a function of , we use the chain rule:
Step 2: Apply the differentiation terms
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For :
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For (product rule):
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For :
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For the constant :
Substituting these into the equation gives:
Step 3: Simplify the first derivative equation
Expand and simplify:
Combine terms involving :
Solve for :
Step 4: Differentiate again to find
Differentiate again with respect to :
Using the quotient rule:
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Differentiate :
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Differentiate :
Substitute these results into the formula:
Step 5: Substitute into the expression
Substitute into the equation. After substituting and simplifying, this gives the expression for in terms of and .
Would you like to proceed with further simplification or stop here? Let me know!
Related Questions:
- How can implicit differentiation be used in practical applications?
- What is the significance of the second derivative in this context?
- Can you demonstrate simplifying the second derivative expression further?
- How does the chain rule apply in implicit differentiation?
- Can we verify the solution by plugging in specific values for and ?
Tip: Always verify before proceeding to find , as simplification can lead to errors if not carefully handled.
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Quotient Rule
Second Derivatives
Formulas
Chain rule: d/dx[f(y)] = f'(y) * dy/dx
Product rule: d/dx[uv] = u'v + uv'
Quotient rule: d/dx[u/v] = (v*u' - u*v') / v²
Theorems
Differentiation Rules
Implicit Differentiation Principles
Suitable Grade Level
Grades 11-12 or early college
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