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GATE DA 2025 Q24: Which combinations of f and g are always differentiable at c?

Question

Consider two functions f:R→Rf : \mathbb{R} \rightarrow \mathbb{R} and g:R→(1,∞)g : \mathbb{R} \rightarrow (1, \infty). Both functions are differentiable at a point cc. Which of the following functions is/are ALWAYS differentiable at cc? The symbol ⋅\cdot denotes product and the symbol ∘\circ denotes composition of functions.

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Answer: A, B, C (official GATE 2025 final answer key)

Concept used: Sum, product and quotient rules, and the condition for the chain rule.

First step: Sums, products and quotients (with a non-zero denominator, and g>1g > 1 here) of functions differentiable at cc are differentiable at cc. For f∘gf \circ g, you would need ff differentiable at g(c)g(c), which is not given.

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Source: official GATE 2025 DA question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key

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