Question

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

Answer 1

`-2/x^3`

Explanation:

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

`= lim_(h→0) ( 1/(x+h)^2 -1/x^2)/h`

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

`= lim_(h→0)(( x^2 - x^2 - 2hx - h^2)/(x^2(x+h)^2))/h`

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

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

`=( - 2x)/x^4 = -2/x^3`