Math Problem Statement
Given the equation
StartRoot x squared plus 6 xy plus y Superscript 6 EndRoot equals 2x2+6xy+y6=2,
evaluate
StartFraction dy Over dx EndFractiondydx.
Assume that the equation implicitly defines y as a differentiable function of x.
Question content area bottom
Part 1
If
F(x,y)equals=StartRoot x squared plus 6 xy plus y Superscript 6 EndRoot minus 2 equals 0x2+6xy+y6−2=0,
then
Upper F Subscript xFxequals=StartFraction nothing Over nothing EndFraction .
Solution
We are tasked with finding the derivative assuming that the equation defines implicitly as a function of . Let's break this down step by step.
Step 1: Understand the given equation
We are given the equation:
We can square both sides to eliminate the square root:
Now, let's define the function as:
This equation implicitly defines as a function of . We are tasked with finding the partial derivative of with respect to , which will be needed for implicit differentiation.
Step 2: Partial derivatives of
We need to compute the partial derivatives and , where is the partial derivative of with respect to , and is the partial derivative with respect to .
Finding :
Differentiate with respect to while treating as a function of :
Applying the chain rule, we get:
Finding :
Now, let's differentiate with respect to :
This gives:
Step 3: Implicit differentiation to find
Using implicit differentiation, we know that:
Substitute the values for and :
Solve for :
Now, divide both sides by :
Final answer:
Would you like to dive deeper into any step or have more questions on this?
Here are five related questions to expand your understanding:
- How do we apply implicit differentiation to more complex equations?
- What are partial derivatives, and how are they useful in multivariable calculus?
- How does squaring both sides of the equation help in solving the problem?
- Why is it important to treat as a function of during differentiation?
- Can we solve for explicitly from the original equation?
Tip: Always check if squaring an equation introduces extraneous solutions!
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Multivariable Calculus
Partial Derivatives
Formulas
sqrt(x^2 + 6xy + y^6) = 2
Implicit Differentiation: Fx + Fy(dy/dx) = 0
dy/dx = -(2x + 6y) / (6x + 6y^5)
Theorems
Chain Rule for Implicit Differentiation
Partial Derivatives
Suitable Grade Level
Grades 11-12, College Calculus
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