GATE DA 2025 Set 1 — Question 14
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Calculus & Optimization → Continuity & Differentiability → Differentiation
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Question
Let . Let denote the derivative of evaluated at .
What is the value of ? (Note: ! denotes factorial)
What is the value of ? (Note: ! denotes factorial)
Correct answer
(A) 0
Solution
The given function is , which is the definition of the hyperbolic sine function, .The derivatives of cycle as follows:
In general:
if is even.
if is odd.We need to find the derivative at , i.e., .
Since is even, .Evaluating at :
.Alternatively, using the Maclaurin series expansion for and :
This series contains only odd powers of . The coefficient of is 0. Since the coefficient of in the Maclaurin series is , we have:
.
In general:
if is even.
if is odd.We need to find the derivative at , i.e., .
Since is even, .Evaluating at :
.Alternatively, using the Maclaurin series expansion for and :
This series contains only odd powers of . The coefficient of is 0. Since the coefficient of in the Maclaurin series is , we have:
.
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