Math Problem Statement
Реши задание
Solution
To solve the given problem, we need to find the derivative of the implicitly defined function , based on the given equation. Let's carefully analyze the problem:
Given equation:
We're tasked to:
- Differentiate the equation implicitly with respect to .
- Use the given point , to evaluate .
Step 1: Implicit differentiation
Differentiating both sides of the equation with respect to , applying the chain rule as necessary:
First term: since the derivative of is .
Second term: using the product rule.
Right-hand side:
Thus, the differentiated equation becomes:
Step 2: Solve for
Rearrange to isolate : Factor out : Solve for :
Step 3: Substitute ,
Substitute and into the equation.
Numerator:
Denominator: First, calculate : So:
Result:
Step 4: Simplify
Simplify the numerator:
Divide by 3:
Factor out :
Upon evaluating, the correct answer corresponds to .
Let me know if you'd like a breakdown or further clarifications!
Related Questions:
- What is implicit differentiation, and why is it needed for this problem?
- How is the chain rule applied when differentiating functions of with respect to ?
- Why do we substitute specific values of and after finding ?
- How does the derivative of influence the calculations in this problem?
- Can the solution be checked by solving for explicitly (if possible)?
Tip:
Always simplify square roots and fractions carefully to avoid mistakes during implicit differentiation!
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Trigonometric Derivatives
Algebraic Manipulation
Formulas
Derivative of \( \arcsin(y) \): \( \frac{d}{dx} \arcsin(y) = \frac{1}{\sqrt{1 - y^2}} \cdot y' \)
Product Rule: \( \frac{d}{dx} (uv) = u'v + uv' \)
Theorems
Chain Rule
Product Rule
Suitable Grade Level
Grades 10-12
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