Math Problem Statement
The equation 4(xy^2+y^2)^(1/4)+xy−20=0
defines the variable y as a function of variable x (y=f(x) ) in the neighborhood of the point (0,5) . Evaluate y′(0) . remember to use the product rule and the chain rule
Solution
To find , we need to implicitly differentiate the given equation with respect to and then solve for at and .
The equation is:
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Differentiate each term with respect to :
Using the product rule, chain rule, and implicit differentiation where :
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First term: Simplifying,
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Second term:
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Third term: Constant term
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Combine derivatives:
Substitute the differentiated terms into the equation:
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Evaluate at and :
Substitute , , and into the equation: Simplifying,
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Solve for :
Since , we have: Multiply by 125:
So, the value of is:
Would you like more details on any specific steps, or have any questions?
Follow-up Questions
- How would the process differ if we needed to find ?
- Can we analyze the behavior of around using this derivative?
- What if the equation were modified to ; how would that affect the solution?
- How does implicit differentiation help in finding derivatives for complex equations?
- Can you explore the significance of in terms of the graph of ?
Tip
Using implicit differentiation is particularly useful for equations where solving explicitly for in terms of is challenging or impossible.
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Product Rule
Chain Rule
Calculus
Algebra
Formulas
Product rule: (uv)' = u'v + uv'
Chain rule: d/dx [f(g(x))] = f'(g(x)) * g'(x)
Theorems
Implicit Differentiation
Rules of Derivatives
Suitable Grade Level
Undergraduate Calculus (first-year college)
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