Given the function fx squarerootx 1 find the difference quo

Given the function f(x)= squareroot(x + 1) find the difference quotient. Simplify.

Solution

difference quotient is

(f(x+h) - f(x))/h

f(x+h)=sqrt(x+h+1)

so we get

(sqrt(x+h+1) - sqrt(x+1))/h

multiplying both top and bottom by sqrt(x+h+1)+ sqrt(x+1)

we get

(x+h+1-x-1)/h(sqrt(x+h+1) +sqrt(x+1)) = 1/(sqrt(x+h+1)+sqrt(x+1))