Using the limit definition, how do you find the derivative of # f(x)=sqrt(x−3)#?

1 Answer
Feb 9, 2016

see explanation

Explanation:

f'(x) = #lim_(h→0)( f(x+h) -f(x))/h#

# =lim_(h→0) (sqrt(x+h-3) - sqrt(x-3))/h#

the aim here is to eliminate h from the denominator so that there is no division by zero as h→ 0.

consider multiplying numerator/denominator by

#sqrt(x+h-3) + sqrt(x-3) #
#color(black)("----------------------------------------------------------")#
numerator# = (sqrt(x+h-3) -sqrt(x-3))(sqrt(x+h-3) + sqrt(x-3))#
distribute brackets using FOIL

# = x+h-3 +sqrt((x-3))sqrt(x+h-3)) -sqrt(x-3)sqrt(x+h-3) -(x-3)#

#=x+h-3-x+3 =h#
#color(black)("-------------------------------------------------------")#
denominator

#sqrt(x-h+3) + sqrt(x-3)#
#color(black)("------------------------------------------------")#
now returning to f'(x)

f'(x) = #lim_(h→0) h/(h(sqrt(x-h+3) + sqrt(x-3))#
#= lim_(h→0) cancel(h)/(cancel(h)sqrt(x-h+3) + sqrt(x-3)#

# = 1/(sqrt(x-3) +sqrt(x-3)) = 1/(2sqrt(x-3))#