How can I solve this problem?Please,help.

Question

#d/(dx)(1/(sqrt(x+1)+sqrt(x+2)))=?#

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Answer 1 · 2017-12-21T15:30:13

#d/dx(1/(sqrt(x+1)+sqrt(x+2)))#

=#1/2(x+2)^(-1/2)-1/2(x+1)^(-1/2)#

#1/(sqrt(x+1)+sqrt(x+2))#

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

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

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

Hence,

#d/dx(1/(sqrt(x+1)+sqrt(x+2)))#

=#d/dx(sqrt(x+2)-sqrt(x+1))#

=#1/2(x+2)^(-1/2)-1/2(x+1)^(-1/2)#

Answer 2 · 2017-12-21T15:36:59

See below.

Rationalizing before differentiation

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

and then

#d/(dx)1/(sqrt(x+1)+sqrt(x+2)) = d/(dx)sqrt(x+2)-d/(dx)sqrt(x+1)= 1/2(1/sqrt(x+2)-1/sqrt(x+1))#