Math Problem Statement
get y'
Solution
To find for the given equation at the point , we'll differentiate the equation implicitly with respect to .
Step-by-Step Solution:
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Given Equation:
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Differentiate Implicitly with respect to : Using the product rule for and the chain rule for :
The derivatives are:
- ,
- ,
- .
Substituting these, we get:
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Simplify and Solve for : Combine all terms involving :
Factor out :
Solve for :
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Substitute : At , substitute and :
Simplify:
Final Answer:
Would you like me to explain any step in more detail?
Related Questions:
- What is implicit differentiation, and why is it useful?
- Can this method be applied to equations without explicit terms?
- What would happen if the point given were not on the curve?
- How does the product rule work for terms like ?
- What is the geometrical interpretation of at a specific point?
Tip:
When solving for in implicit differentiation, always isolate the terms with carefully to avoid errors in factoring.
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Derivatives
Product Rule
Chain Rule
Formulas
Implicit differentiation: d/dx[f(x, y)] = 0
Product Rule: d/dx[uv] = u'v + uv'
Chain Rule: d/dx[g(y)] = g'(y) * dy/dx
Theorems
Rules of Differentiation
Implicit Differentiation Theorem
Suitable Grade Level
Grades 11-12, College Level Calculus
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