How do you use the definition of a derivative to find f' given f(x)=1/x^2 at x=1?

1 Answer
Jan 16, 2018

-2

Explanation:

f(x) = 1/x^2

The derivative is defined as:

lim_(h->0)(f(x+h)-f(x))/h

Which in our case would be:

=lim_(h->0)(1/(x+h)^2-1/x^2)/h=lim_(h->0)(((x^2-(x+h)^2)/(x^2(x+h)^2)))/h

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

Substitute x=1:

=lim_(h->0)(1^2-(1+h)^2)/(1^2(1+h)^2h)

=lim_(h->0)(1-(1+h)^2)/((1+h)^2h)=lim_(h->0)(1-1-2h-h^2)/((1+h)^2h)

lim_(h->0)(-2h-h^2)/((1+h)^2h)=lim_(h->0)(-(2+h)h)/((1+h)^2h)

lim_(h->0)(-(2+h))/((1+h)^2)

Evaluate the limit:

=-2/1=-2