GATE DA 2025 Set 1 — Question 24
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Calculus & Optimization → Continuity & Differentiability → Differentiability
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Question
Consider two functions and . Both functions are differentiable at a point . Which of the following functions is/are ALWAYS differentiable at ? The symbol denotes product and the symbol denotes composition of functions.
Correct answer
(A) f ± g; (B) f · g; (C) (f)/(g)
Solution
Let and be differentiable at .
(A) : The sum or difference of two differentiable functions is differentiable at that point. Correct.
(B) : The product of two differentiable functions is differentiable at that point (Product Rule). Correct.
(C) : The quotient of two differentiable functions is differentiable at provided . Since the codomain of is , , so . Correct.
(D) : The chain rule states that for to be differentiable at , must be differentiable at AND must be differentiable at . We are only given that is differentiable at , not necessarily at (unless , which is not guaranteed). Similarly for , we need to be differentiable at , which is not guaranteed. Thus, this is not ALWAYS differentiable. Incorrect.
(A) : The sum or difference of two differentiable functions is differentiable at that point. Correct.
(B) : The product of two differentiable functions is differentiable at that point (Product Rule). Correct.
(C) : The quotient of two differentiable functions is differentiable at provided . Since the codomain of is , , so . Correct.
(D) : The chain rule states that for to be differentiable at , must be differentiable at AND must be differentiable at . We are only given that is differentiable at , not necessarily at (unless , which is not guaranteed). Similarly for , we need to be differentiable at , which is not guaranteed. Thus, this is not ALWAYS differentiable. Incorrect.
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