We use the formula f'(x) = lim_(h->0) (f(x+ h) - f(x))/h to find the derivative.
f'(x) = lim_(h->0) (sqrt(x + h +1) - sqrt(x + 1))/h
Multiply this by the conjugate of the numerator, which is sqrt(x+ h+ 1) + sqrt(x + 1).
f'(x) = lim_(h->0) (sqrt(x + h + 1) - sqrt(x + 1))/h * (sqrt(x + h + 1) + sqrt(x + 1))/(sqrt(x + h + 1) + sqrt(x + 1))
f'(x) = lim_(h-> 0) (x+ h + 1 - (x + 1))/(hsqrt(x + h +1) + hsqrt(x + 1))
f'(x) = lim_(h->0) (x + h + 1- x - 1)/(h(sqrt(x + h + 1) + sqrt(x + 1)))
f'(x) = lim_(h->0) h/(h(sqrt(x + h + 1) + sqrt(x +1))
f'(x) = lim_(h->0) 1/(sqrt(x + h + 1) + sqrt(x + 1))
You can now evaluate through substitution.
f'(x) = 1/(sqrt(x + 0 + 1) + sqrt(x + 1))
f'(x) = 1/(2sqrt(x+ 1))
Hopefully this helps!
