CBSE Class 12 Mathematics 2026 Main Paper 65/4/3 · Q313 Marksimplicit_derivatives
CBSE Class 12 Mathematics 2026 Main Paper 65/4/3 · Question 31 (Continuity And Differentiability)
If \(xy=e^{x-y}\), find dy/dx. OR Differentiate \(\tan^{-1}\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\) with respect to \(\cos^{-1}(x^2)\).
Alternative 1 3 Marks
If \(xy=e^{x-y}\), find dy/dx.
Answer: (e^(x−y)−y)/(x+e^(x−y)), where the denominator is nonzero.
- Differentiate both sides: y+xy′=e^(x−y)(1−y′).
- Collect the y′ terms on the left.
- Divide by x+e^(x−y).
Alternative 2 3 Marks
Differentiate \(\tan^{-1}\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\) with respect to \(\cos^{-1}(x^2)\).
Answer: −1/2.
- Set θ=arccos(x²) in [0,π/2]. The square roots become √2 cos(θ/2) and √2 sin(θ/2).
- The ratio is tan(π/4−θ/2), whose angle is principal.
- The expression is π/4−θ/2, with derivative −1/2 in θ.