Question

How do you find the derivative of `f(x)=1/x` using the limit definition?

Answer 1

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

Explanation:

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

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

Resolve the numerator into one fraction:

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

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