Let f: ℝ-{0} → ℝ be a function satisfying f(x/y)=f(x)/f(y) for all x, y, f(y) ≠ 0. If f′(1)=2024, then

Solution

Maths sir · 4 steps

Differentiate with respect to x keeping y fixed, then put x = y so that f'(1) appears.

1. Find f(1). Put x = y in the equation. The left side becomes f(1) and the right side is f(y)/f(y) = 1, so f(1) = 1.
2. Differentiate in x. Treat y as a constant and differentiate both sides with respect to x. On the left the chain rule gives an extra factor 1/y. On the right f(y) is only a constant.

   1/y f'(x/y) = f'(x)/f(y)

3. Put x = y. Now f'(x/y) becomes f'(1) = 2024. This gives 2024/y = f'(y)/f(y), that is, y f'(y) = 2024 f(y) for every y ≠ 0.
4. So. Rename y as x: x f'(x) - 2024 f(x) = 0, which is option (c). As a check, f(x) = x²⁰²⁴ satisfies the equation and has f'(1) = 2024.

The idea to keep. For a functional equation with a known derivative at one point, differentiate with respect to one variable, then choose the other so the known point appears. Here that point is x = y.

Answer from the NTA final answer key. Solution by PrepWiser.