Math Problem Statement
Solution
Let's solve the given problem.
We are tasked with differentiating the equation implicitly to find in terms of and .
Step 1: Differentiate both sides
The equation is:
Differentiating term by term with respect to , we apply the chain rule for , as is a function of :
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The derivative of is:
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The derivative of is:
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The derivative of is:
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The derivative of is simply:
Step 2: Combine the derivatives
Now substitute these derivatives into the equation:
Step 3: Factorize terms involving
Group terms with on one side:
Solve for :
Final Answer:
Would you like a further breakdown or related examples?
Related Questions:
- How do we apply the chain rule in implicit differentiation?
- Why do we need to treat as a function of ?
- How would the solution change if the equation was ?
- Can implicit differentiation be used to find higher-order derivatives?
- What does represent geometrically in this case?
Tip:
Always group terms with together before isolating it—this simplifies solving!
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Chain Rule
Differentiation
Formulas
dy/dx = -4x / (8y + 1)
Theorems
Chain Rule for Implicit Differentiation
Suitable Grade Level
Grades 11-12
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