Given f(x)=(x)/(x+1)
Find....
(f(x+h)-f(x))/h
To find the expression (f(x+h) - f(x))/h for the given function f(x) = x/(x+1), we need to first substitute (x+h) in place of x in the function, and then subtract the initial function f(x) from it. Finally, we divide the result by h.
Step 1: Substitute (x+h) in place of x in the function:
f(x+h) = (x+h)/((x+h)+1)
Step 2: Find f(x):
f(x) = x/(x+1)
Step 3: Subtract f(x) from f(x+h):
f(x+h) - f(x) = [(x+h)/((x+h)+1)] - [x/(x+1)]
Step 4: Simplify the expression:
To simplify the expression, we need to find the least common denominator (LCD) of the two fractions in the numerator. The LCD in this case is (x+1)(x+h+1).
[(x+h)(x+1) - x(x+h+1)] / [(x+1)(x+h+1)]
= [x^2 + hx + x + h - x^2 - hx - x] / [(x+1)(x+h+1)]
= [h] / [(x+1)(x+h+1)]
Step 5: Divide the result by h:
(f(x+h) - f(x))/h = [h] / [h(x+1)(x+h+1)]
= 1 / [(x+1)(x+h+1)]
Therefore, the expression (f(x+h) - f(x))/h for the given function f(x) = x/(x+1) is 1 / [(x+1)(x+h+1)].