Question

How do you use the limit definition of the derivative to find the derivative of `f(x)=1/(x+2)`?

Answer 1

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

Explanation:

Use `f'(x) = lim_(h->0) (f(x + h) - f(x))/h`.

`f'(x) = lim_(h->0) (1/(x + h + 2) - 1/(x + 2))/h`

`f'(x) = lim_(h->0) ((x + 2)/((x + h + 2)(x + 2)) - (x + h + 2)/((x + 2)(x + h + 2)))/h`

`f'(x) = lim_(h-> 0) ((x + 2 - x - h - 2)/((x + h + 2)(x + 2)))/h`

`f'(x) = lim_(h->0) (-h)/((x+ h + 2)(x + 2)h)`

`f'(x) = lim_(h->0) -1/((x + h + 2)(x + 2))`

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

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

Verification with the chain rule and the quotient rule yield the same result.

Hopefully this helps!