NCERT Class 12 · Ch 5 Exercise 5.3 · Q6
NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.3 Question 6
Find dy/dx for \(x^{3} + x^{2} y + x y^{2} + y^{3} - 81 = 0\).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: \(- \frac{3 x^{2} + 2 x y + y^{2}}{x^{2} + 2 x y + 3 y^{2}}\) where \(x^{2} + 2 x y + 3 y^{2}\)≠0.
- Differentiate the relation treating y as a function of x. The partial derivatives are Fₓ=\(3 x^{2} + 2 x y + y^{2}\) and Fᵧ=\(x^{2} + 2 x y + 3 y^{2}\).
- The chain rule gives Fₓ+Fᵧy′=0. Thus \(x^{2} + 2 x y + 3 y^{2}\)y′ = \(- 3 x^{2} - 2 x y - y^{2}\).
- At points where Fᵧ≠0, divide to obtain y′=\(- \frac{3 x^{2} + 2 x y + y^{2}}{x^{2} + 2 x y + 3 y^{2}}\). Points with Fᵧ=0 require a separate local-curve check.
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