Question

How do you use the definition of a derivative to find the derivative of `1/x^2`?

Answer 1

We use the formula `f'(x) = lim_(h -> 0) (f(x + h) - f(x))/h` for "definition of derivative" problems.

Explanation:

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

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

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

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

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

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

We can now evaluate:

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

`f'(x) = (-2x)/x^4`

`f'(x) = -2/x^3`

Hopefully this helps!