How to calculate using derivative definition

Question

I am trying to find the derivative of $f$ at $x=1$ when $y=\sqrt{3x+1}$ using only the following derivative definition:

$$f^\prime(1)=\lim_{x\to1}\frac{f(x)−f(1)}{x−1}$$

I think I need to get rid of the radical first, but I simply cannot find the right way. I am relearning math after many years, so please bear with me.

Answer 1

Since $(\sqrt{3x+1}-2)(\sqrt{3x+1}+2)=3(x-1)$,$$f^\prime(1)=\lim_{x\to1}\frac{\sqrt{3x+1}-2}{x-1}=\lim_{x\to1}\frac{3}{\sqrt{3x+1}+2}=\frac34.$$

Answer 2

Hint: $\displaystyle\sqrt b-\sqrt a=\frac{b-a}{\sqrt b+\sqrt a}$.