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

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Answer 1 · 2018-01-16T18:07:10

#-2#

#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#