Question

How do you find the derivative using limits of `f(x)=1/x^2`?

Answer 1

`lim_(Deltax->0) (1/(x+Deltax)^2-1/x^2)/(Deltax) = -2/x^3`

Explanation:

The limit definition of the derivative of a function `f(x)` is:

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

Let's calculate the increment of the function between `x` and `x+Deltax`:

`Delta f = 1/(x+Deltax)^2 -1/x^2 = (x^2 - (x+Deltax)^2)/(x^2(x+Deltax)^2) = (x^2 - x^2 -2xDeltax -(Deltax)^2)/(x^2(x+Deltax)^2) = (-2xDeltax -(Deltax)^2)/(x^2(x+Deltax)^2)`

The incremental ratio is then:

`(Deltaf)/(Deltax) =(-2x -Deltax)/(x^2(x+Deltax)^2)`

and passing to the limit:

`lim_(Deltax->0) (-2x -Deltax)/(x^2(x+Deltax)^2) = -2x/(x^2*x^2) = -2/x^3`

Answer 2

`f'(x)=color(green)(-2 * 1/x^3)`
`color(white)("XXXXXX")`(see below for determination using limits)

Explanation:

Using the limit definition for a derivative:
`color(white)("XXX")f'(x)=lim_(hrarr0)(f(x+h)-f(x))/h`

For the case `f(x)=1/x^2`

`f(x+h)-f(x) = 1/(x+h)^2-1/x^2`

`color(white)("XXXXXXXXXX")=(x^2 - (x+h)^2)/(x^2 * (x+h)^2)`

`color(white)("XXXXXXXXXX")=(-2xh-h^2)/(x^4+2hx^3+h^2x^2)`

and theerfore
`(f(x+h)-f(x))/h = (-2x-h)/(x^4+2hx^3+h^2x^2)`

This is defined when `h=0`
so
`lim_(hrarr0) (f(x+h)-f(x))/h=(-2x-0)/(x^4+2 * 0 * x^3 + 0^2 * x^2)`

`color(white)("XXXXXXXXXXXXX")=(-2x)/(x^4) = -2/x^3=(-2) * (1/x^3)`