Math Problem Statement
Solution
To solve the problem, we need to find using implicit differentiation for the given equation:
Step 1: Differentiate both sides with respect to .
Using implicit differentiation, differentiate with respect to , while considering that is a function of :
Now, find using the quotient rule:
So, substituting this back, we get:
Step 2: Solve for .
Since , we can equate the numerator to zero:
This simplifies to:
Thus,
The solution is:
Would you like further details or have any other questions?
Related Questions:
- What is implicit differentiation, and how does it differ from explicit differentiation?
- How can we apply the chain rule in implicit differentiation?
- What is the quotient rule, and how is it used in differentiation?
- Why do we need to differentiate both sides of the equation with respect to ?
- How can you solve implicit differentiation problems where multiple terms contain ?
Tip:
When dealing with implicit differentiation, always remember to treat as a function of , applying the chain rule appropriately.
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Trigonometric Functions
Quotient Rule
Chain Rule
Formulas
sin(x/y) = 1/2
Quotient rule: (d/dx)(u/v) = (v * du/dx - u * dv/dx) / v^2
Chain rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)
Theorems
Differentiation Theorem
Trigonometric Derivatives
Suitable Grade Level
College-level Calculus (First-year)
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