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