Class 12 Continuity And Differentiability · Long Answer3 Markssecond_derivatives
Class 12 Maths · Continuity And Differentiability · Long Answer #96
For y=x^x, x>0, prove y″−(y′)²/y−y/x=0.
✅ Step-by-Step Solution
Final Answer: Identity proved.
- Take natural logarithms: ln y=x ln x.
- Differentiate: y′/y=1+ln x, so y′=y(1+ln x).
- Differentiate again: y″=y′(1+ln x)+y/x=(y′)²/y+y/x, using the first-derivative relation.
- Rearranging proves y″−(y′)²/y−y/x=0.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Take natural logarithms: ln y=x ln x.
- [1 mark] Differentiate: y′/y=1+ln x, so y′=y(1+ln x).
- [1 mark] Differentiate again: y″=y′(1+ln x)+y/x=(y′)²/y+y/x, using the first-derivative relation.
- [0.5 marks] Rearranging proves y″−(y′)²/y−y/x=0.
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