Question

How do you find the derivative of `1/x^2`?

Answer 1

`-2/x^3`

Explanation:

We will use the power rule, which states that the derivative of `x^n` is `nx^(n-1)`.

We can use the power rule once we write `1/x^2` as `x^-2`.

Thus, according to the power rule, the derivative of `x^-2` is `-2x^(-2-1)=-2x^-3=-2/x^3`.

Answer 2

Use the limit definition to find:

`d/(dx) 1/x^2 = -2/x^3`

Explanation:

The power rule is good, but let's find it using the limit definition:

Let `f(x) = 1/x^2`

Then:

`d/(dx) 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)/(h(x+h)^2x^2)`

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

`= lim_(h->0)(-color(red)(cancel(color(black)(h)))(2x+h))/(color(red)(cancel(color(black)(h)))(x+h)^2x^2)`

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

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

`= -2/x^3`