Question

How do you find f'(x) using the definition of a derivative for `f(x)=1/x^2`?

Answer 1

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

Explanation:

`f(x) = 1/x^2`
`f'(x) = lim_(h \rarr 0) (f(x+h) - f(x))/h`
`= lim_(h \rarr 0) (1/(x+h)^2 - 1/x^2)/h`
`= lim_(h \rarr 0) (x^2 - (x+h)^2)/(h(x+h)^2x^2)`
`= lim_(h \rarr 0) (-2xh-h^2)/(h(x+h)^2x^2)`
`= lim_(h \rarr 0) (-2x-h)/((x+h)^2x^2)`
`= -2/x^3`