Q(s) = (f(x+s) - f(x-s))/(2s)
To simplify this expression for the function f(t) = t/(t+1), we substitute f(t) with its given expression:
Q(s) = ((x+s)/(x+s+1) - (x-s)/(x-s+1))/(2s)
Next, we simplify the numerator:
Q(s) = ((x+s)(x-s+1) - (x-s)(x+s+1))/(2s(x+s+1)(x-s+1))
Expanding the numerator:
Q(s) = (x^2 - xs + x + xs + s^2 + s - x^2 + xs + x - xs - s^2 + s)/(2s(x+s+1)(x-s+1))
Simplifying further:
Q(s) = (2s)/(2s(x+s+1)(x-s+1))
Finally, canceling out the common factors:
Q(s) = 1/(x+s+1)(x-s+1)