Question

How do you find f'(2) using the limit definition given `3/(x+1)`?

Answer 1

The limit definition at a point is that the derivative of `f(x)` at `x=a` is:

`f'(a)=lim_(xrarra)(f(x)-f(a))/(x-a)`

So here, where `f(x)=3/(x+1)` and `a=2`:

`f'(2)=lim_(xrarr2)(3/(x+1)-3/(2+1))/(x-2)`

`f'(2)=lim_(xrarr2)(3/(x+1)-1)/(x-2)`

`f'(2)=lim_(xrarr2)(3/(x+1)-(x+1)/(x+1))/(x-2)`

`f'(2)=lim_(xrarr2)((3-(x+1))/(x+1))/(x-2)`

`f'(2)=lim_(xrarr2)(3-(x+1))/(x+1)(1/(x-2))`

`f'(2)=lim_(xrarr2)(-x+2)/(x+1)(1/(x-2))`

`f'(2)=lim_(xrarr2)(-(x-2))/(x+1)(1/(x-2))`

`f'(2)=lim_(xrarr2)(-1)/(x+1)`

`f'(2)=(-1)/(2+1)`

`f'(2)=-1/3`