CBSE Class 12 Mathematics 2026 Main Paper 65/5/1 · Q335 Marksimplicit_derivatives
CBSE Class 12 Mathematics 2026 Main Paper 65/5/1 · Question 33 (Continuity And Differentiability)
If \(y\sqrt{x^2+1}=\ln(\sqrt{x^2+1}-x)\), show that \((x^2+1)y\prime+xy+1=0\). OR Differentiate \(x^{\cot x}+(2x^2-3)/(2x^2-x+2)\).
Alternative 1 5 Marks
If \(y\sqrt{x^2+1}=\ln(\sqrt{x^2+1}-x)\), show that \((x^2+1)y\prime+xy+1=0\).
Answer: (x²+1)y′+xy+1=0.
- Write u=√(x²+1), so u′=x/u and u−x>0.
- The right side has derivative (u′−1)/(u−x)=−1/u. On the left, the derivative is uy′+xy/u.
- Multiply the equality by u and rearrange.
Alternative 2 5 Marks
Differentiate \(x^{\cot x}+(2x^2-3)/(2x^2-x+2)\).
Answer: x^(cot x)[cot x/x−csc²x ln x]+(−2x²+20x−3)/(2x²−x+2)².
- Logarithmically differentiate x^(cot x), using ln(x^(cot x))=cot x ln x.
- For the rational term apply the quotient rule. Its numerator simplifies to −2x²+20x−3.
- Add the derivatives where the original real expression is defined for x>0 and sin x≠0.