Math Problem Statement
Given that \(2xe^y + 5\cos x = x^3\), find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\).
Solution
To find for the given equation , we differentiate both sides of the equation with respect to , applying implicit differentiation.
Given:
Steps:
-
Differentiate each term:
- For , use the product rule:
- For , the derivative is:
- For , the derivative is:
-
Combine the results:
-
Isolate :
- Move all terms involving to one side:
- Solve for :
Final Answer:
Would you like a step-by-step explanation of any part? Let me know!
Here are five related questions to deepen your understanding:
- What is implicit differentiation, and why is it necessary here?
- How does the product rule apply to ?
- What is the significance of isolating in terms of and ?
- How does the chain rule apply when differentiating ?
- What happens if in the expression for ?
Tip: Always identify terms that require the chain rule when differentiating implicitly.
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Math Problem Analysis
Mathematical Concepts
Calculus
Implicit Differentiation
Formulas
Product Rule: \(\frac{d}{dx}[uv] = u'v + uv'\)
Chain Rule: \(\frac{d}{dx} e^y = e^y \frac{dy}{dx}\)
Theorems
Differentiation Rules
Suitable Grade Level
Undergraduate Calculus