NCERT Class 12 · Ch 5 Exercise 5.5 · Q15
NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.5 Question 15
Find dy/dx if xʸ=e^(x−y).
💡 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: \(\frac{x e^{x} e^{- y} - x^{y} y}{x \left(x^{y} \log{\left(x \right)} + e^{x} e^{- y}\right)}\) where the stated powers and derivative are defined.
- Use logarithmic/chain differentiation of each variable power. Work with x>0; y may be any real value.
- Differentiating with respect to x gives \(- e^{x - y} + \frac{x^{y} y}{x}\) + \(x^{y} \log{\left(x \right)} + e^{x - y}\) y′=0.
- Collect y′ terms and divide by \(x^{y} \log{\left(x \right)} + e^{x - y}\) when nonzero: y′=\(\frac{x e^{x} e^{- y} - x^{y} y}{x \left(x^{y} \log{\left(x \right)} + e^{x} e^{- y}\right)}\).
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