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**zebra2147** I have done 2 of the 3 parts of the following problem but now I'm stuck. Any help would be appreciated:

Let $\displaystyle \mathbb{R}^+=(0,\infty)$. Suppose $\displaystyle f:\mathbb{R}^+\rightarrow \mathbb{R}$ is differentiable at $\displaystyle x=1$ and $\displaystyle f(xy)=f(x)+f(y)$ for all $\displaystyle x,y\in \mathbb{R}^+$.

1)Prove $\displaystyle f(1)=0$

2)Prove $\displaystyle f(1/x)=-f(x)$

3)Prove $\displaystyle f$ is differentiable on $\displaystyle \mathbb{R}^+$, and $\displaystyle f'(x)=f'(1)/x$.

I have already proven parts 1 and 2. For part 3 I'm guessing you set it up something like this... $\displaystyle (f(xy)-f(a))/(xy-a)$ and then you use the fact that $\displaystyle f(xy)=f(x)+f(y)$ to rearrange to find the derivative but I'm struggling to see how. Any help would be great.