CBSE Class 12 Mathematics 2026 Main Paper 65/5/2 · Q252 Markslogarithmic_differentiation
CBSE Class 12 Mathematics 2026 Main Paper 65/5/2 · Question 25 (Continuity And Differentiability)
Differentiate xˣ with respect to x ln x, for x>0. OR If y=P cos(ux)+Q sin(ux), with constants P,Q,u, show that y″+u²y=0.
Alternative 1 2 Marks
Differentiate xˣ with respect to x ln x, for x>0.
Answer: xˣ.
- Set u=x ln x.
- Then xˣ=exp(u).
- Differentiate in u to obtain exp(u)=xˣ.
Alternative 2 2 Marks
If y=P cos(ux)+Q sin(ux), with constants P,Q,u, show that y″+u²y=0.
Answer: y″+u²y=0.
- Differentiate to obtain y′=−Pu sin(ux)+Qu cos(ux).
- Differentiate again to get y″=−Pu² cos(ux)−Qu² sin(ux).
- Factor −u² to get y″=−u²y.