Let f:R-0 R be a function satisfying f((x)/(y))=(f(x))/(f(y)) for all x,y,f(y) 0.

Question

Let $f:R-{0}	o R$ be a function satisfying $f(\frac{x}{y})=\frac{f(x)}{f(y)}$ for all $x,y,f(y)
e 0$. If ${f}^{'}(1)=2024$, then

Accepted Answer

$$f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}$$

$$\begin{aligned}
& \mathrm{f}^{\prime}(1)=2024 \
& \mathrm{f}(1)=1
\end{aligned}$$

Partially differentiating w. r. t. x

$$\begin{aligned}
& \mathrm{f}^{\prime}\left(\frac{\mathrm{x}}{\mathrm{y}}\right) \cdot \frac{1}{\mathrm{y}}=\frac{1}{\mathrm{f}(\mathrm{y})} \mathrm{f}^{\prime}(\mathrm{x}) \
& \mathrm{y} \rightarrow \mathrm{x} \
& \mathrm{f}^{\prime}(1) \cdot \frac{1}{\mathrm{x}}=\frac{\mathrm{f}^{\prime}(\mathrm{x})}{\mathrm{f}(\mathrm{x})} \
& 2024 \mathrm{f}(\mathrm{x})=\mathrm{xf}^{\prime}(\mathrm{x}) \Rightarrow \mathrm{xf}^{\prime}(\mathrm{x})-2024 \mathrm{~f}(\mathrm{x})=0
\end{aligned}$$

Solutions · 1