Question

Find the derivative by definition of
F(x)= x/x+1

Answer 1

by first principles:
f(x) = x/(x+1), notice the necessary brackets
f(x+h) = (x+h)/(x+h + 1)

F'(x) = limit ((x+h)/(x+h + 1) - x/(x+1) )/(x+h - x), as h ----> 0
= lim [( (x+h)(x+1) - x(x+h + 1) ) / ((x+1)(x+h+1) ]/h , as h ---> 0
= lim [ (x^2 + hx + x + h - x^2 - xh - x)/((x+1)(x+h+1) ] / h , as h ---> 0
= lim [ h/((x+1)(x+h+1)]/h , as h ---> 0
= 1/((x+1)(x+h+1) , as h ---> 0
= 1/(x+1)^2

Answer 2

Thank you!