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CBSE Class 12 Mathematics 2026 Main Paper 65/5/1 · Q222 Markslogarithmic_differentiation

CBSE Class 12 Mathematics 2026 Main Paper 65/5/1 · Question 22 (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.
  1. Set u=x ln x.
  2. Then xˣ=exp(u).
  3. 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.
  1. Differentiate to obtain y′=−Pu sin(ux)+Qu cos(ux).
  2. Differentiate again to get y″=−Pu² cos(ux)−Qu² sin(ux).
  3. Factor −u² to get y″=−u²y.

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