Math Problem Statement
Find d/dx (y) if y ^ 2 - x ^ 2 = 2
Solution
Given the equation:
We need to find , or , using implicit differentiation. Let's proceed step by step:
Step 1: Differentiate both sides with respect to .
Differentiate the left-hand side term by term.
Using the chain rule:
- For , treat as a function of (since ):
- For :
On the right-hand side, the derivative of the constant is 0:
Step 2: Set up the equation.
After differentiating, we get:
Step 3: Solve for .
First, isolate the term with :
Now, divide both sides by :
Thus, the derivative is:
Would you like more details or have any questions?
Here are 5 related questions for deeper understanding:
- How would you find (the second derivative)?
- What happens if you substitute specific values for and ?
- How would you interpret the result graphically?
- Can you use implicit differentiation if the equation involves higher powers of ?
- What is the physical interpretation of in real-world scenarios?
Tip: Implicit differentiation is useful when it’s difficult or impossible to solve for one variable explicitly before differentiating.
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Derivatives
Chain Rule
Formulas
d/dx(y^2) = 2y * dy/dx
d/dx(x^2) = 2x
Theorems
Chain Rule of Differentiation
Suitable Grade Level
Grades 10-12
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