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GATE 2025 DA – Question 24

Calculus and Optimization · Limits, continuity and differentiability · 1 mark · Multiple select

Consider two functions $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to (1, \infty)$. Both functions are differentiable at a point $c$. Which of the following functions is/are ALWAYS differentiable at $c$? The symbol $\cdot$ denotes product and the symbol $\circ$ denotes composition of functions.

  1. $f \pm g$
  2. $f \cdot g$
  3. $\frac{f}{g}$
  4. $f \circ g + g \circ f$

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Correct answer: (A) $f \pm g$; (B) $f \cdot g$; (C) $\frac{f}{g}$

Explanation

Sums, differences and products of functions differentiable at $c$ are differentiable at $c$ (A and B). The quotient $\frac{f}{g}$ is differentiable at $c$ because $g$ is differentiable at $c$ and $g > 1$, so it never equals zero (C). For the composition $f \circ g$ the chain rule needs $f$ to be differentiable at $g(c)$, and for $g \circ f$ it needs $g$ to be differentiable at $f(c)$. We are told only about differentiability at the point $c$, so these are not guaranteed, and D need not be differentiable.