# How do you use the limit definition to find the derivative of #f(x)=1/x^2#?

##### 1 Answer

Jul 15, 2016

The limit definition of the derivative tells us that

Since

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

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

#= lim_(h->0) ((cancel(x^2)cancel(-x^2)-2xh-h^2)/(x^2(x^2+2xh+h^2)))/(h)#

#= lim_(h->0) ((cancel(h)(-2x-h))/(x^2(x^2+2xh+h^2))*1/cancel(h))/(cancel(h/1*1/h)#

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

Substituting

#= (-2x-(0))/(x^2(x^2+2x(0)+(0)^2)) = (-2x)/(x^4) = -2/x^3 #

We can check our answer by following the power rule derivatives really quickly: