How do you differentiate #f(x)=x/(1+sqrtx)#?

1 Answer
Nov 25, 2016

#f'(x)= (2+sqrtx)/(2(1+sqrtx)^2#

Explanation:

#f(x)" "# is differentiated by applying the quotient rule
#" "#
differentiation.
#" "#
Quotient Rule:
#" "#
#color(blue)((u/v)'=(u'v - v'u)/v^2#
#" "#
#f'(x)=(x/(1+sqrtx))'#
#" "#
#f'(x)= color(blue)((x'(1+sqrtx)-(1+sqrtx)'x)/(1+sqrtx)^2)#
#" "#
#f'(x)= (1(1+sqrtx)-(0+1/(2sqrtx))x)/(1+sqrtx)^2#
#" "#
#f'(x)= (1+sqrtx-(1/(2sqrtx))x)/(1+sqrtx)^2#
#" "#
#f'(x)= (1+sqrtx-x/(2sqrtx))/(1+sqrtx)^2#
#" "#
#f'(x)= ((2sqrtx + 2x-x)/(2sqrtx))/(1+sqrtx)^2#
#" "#
#f'(x)= ((2sqrtx+x) /(2sqrtx))/(1+sqrtx)^2#
#" "#
#f'(x)= ((sqrtx(2+sqrtx)) /(2sqrtx))/(1+sqrtx)^2#
#" "#
#f'(x)= ((2+sqrtx)/2)/(1+sqrtx)^2#
#" "#
#f'(x)= (2+sqrtx)/(2(1+sqrtx)^2#