By definition:
#(df)/dx = lim_(h->0) ( f(x+h)-f(x))/h#
For #f(x) = sqrt(x+1)# we have:
#d/dx(sqrt(x+1)) = lim_(h->0)(sqrt(x+h+1)-sqrt(x+1))/h#
Multiply and divide the function by #(sqrt(x+h+1)+sqrt(x+1))#:
#d/dx(sqrt(x+1)) = 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)))#
and use the identity: #(a+b)(a-b) = a^2-b^2#
#d/dx(sqrt(x+1)) = lim_(h->0)((cancelx+h+cancel1)-(cancelx+cancel1))/(h(sqrt(x+h+1)+sqrt(x+1))#
#d/dx(sqrt(x+1)) = lim_(h->0)cancelh/(cancelh(sqrt(x+h+1)+sqrt(x+1))#
#d/dx(sqrt(x+1)) = lim_(h->0)1/(sqrt(x+h+1)+sqrt(x+1)) =1/(2sqrt(x+1))#
