Consider two functions f : R -> R and g : R -> R. Both functions are differentiable at a point c. Which of the following functions is/are ALWAYS differentiable at c? The symbol * denotes product and the symbol o denotes composition of functions. A. f * g B. f + g C. f - g D. f o g

GATE 2025 · Engineering Mathematics · Differentiation · medium

Answer: A, B, C

1. Sum and Difference Rules: Since f'(c) and g'(c) both exist, f+g and f-g are differentiable at c. So B (f+g) and C (f-g) are always differentiable at c.
2. Product Rule: Both f'(c) and g'(c) exist, and differentiable functions are continuous so f(c) and g(c) are finite. Hence f*g is differentiable at c. So A (f*g) is always differentiable at c.
3. Composition - Chain Rule requires more: For f o g to be differentiable at c, we need f to be differentiable at g(c). But we are only told f is differentiable at c, not necessarily at g(c). So f o g is NOT always differentiable at c. Option D is excluded.