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

##### 2 Answers

May 16, 2016

#### Explanation:

We will use the power rule, which states that the derivative of

We can use the power rule once we write

Thus, according to the power rule, the derivative of

May 16, 2016

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

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#