Question

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

Answer 1

When you are asked to differentiate using the definition of the derivative, you need to use the formula `f'(x) =lim_(h -> 0) (f(x + h) - f(x))/h`

Explanation:

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

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

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

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

`f'(x) = -1/(x^2 + x xx 0)`

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

Using the power rule to confirm:

`f(x) = x^-1`

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

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

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

Hopefully this helps!