How can I solve this problem?Please,help.

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

2 Answers
Dec 21, 2017

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

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

Explanation:

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)

Dec 21, 2017

See below.

Explanation:

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))