How to use the alternate definition to find the derivative of ##f(x)=sqrt(x+3)## at x=1?
February 20th, 2023
Here are a couple ways you can do the limit calculation for the derivative. Both methods involve “rationalizing the numerator” (not the denominator) as a trick to help you calculate the limits.
##f'(1)=lim_{h->0}frac{f(1+h)-f(1)}{h}##
##=lim_{h->0}frac{sqrt{4+h}-2}{h}cdot frac{sqrt{4+h}+2}{sqrt{4+h}+2}##
##=lim_{h->0}frac{4+h-4}{h(sqrt{4+h}+2)}=lim_{h->0}frac{1}{sqrt{4+h}+2}=frac{1}{4}##
OR
##f'(1)=lim_{x->1}frac{f(x)-f(1)}{x-1}=lim_{x->1}frac{sqrt{x+3}-2}{x-1}##
##=lim_{x->1}frac{x+3-4}{(x-1)(sqrt{x+3}+2)}##
##=lim_{x->1}frac{1}{sqrt{x+3}+2}=frac{1}{4}##