Method of Differentiation
Higher Order Derivatives
IIT-JEE 1985
Grade 12
Question:
If $y = (x + \sqrt{x^2 + 1})^n$, then $(x^2 + 1)y'' + xy'$ equals:
(1) $n^2y$
(2) $-n^2y$
(3) $ny$
(4) $ny'$
Step-by-Step Solution
Key Concept: Differentiate once to show $\sqrt{x^2 + 1} y' = ny$ — this neat intermediate relation is the key step before differentiating a second time.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)