Given:f(x) = 1/(x - 1)f(x)=1x−1, then f(x + h) = 1/(x + h - 1)f(x+h)=1x+h−1
Find f'(x)
f'(x) = lim_(hto0){f(x + h) - f(x)}/h
Substitute 1/(x + h - 1) for f(x + h) and 1/(x - 1) for f(x):
f'(x) = lim_(hto0){1/(x + h - 1) - 1/(x - 1)}/h
Multiply by 1 in the form of (x - 1)/(x - 1):
f'(x) = lim_(hto0){1/(x + h - 1) - 1/(x - 1)}/h(x - 1)/(x - 1)
Multiply numerators and denominators:
f'(x) = lim_(hto0){(x - 1)/(x + h - 1) - (x - 1)/(x - 1)}/(h(x - 1))
The second term in the numerator becomes 1:
f'(x) = lim_(hto0){(x - 1)/(x + h - 1) - 1}/(h(x - 1))
Multiply by 1 in the form of (x + h - 1)/(x + h - 1):
f'(x) = lim_(hto0){(x - 1)/(x + h - 1) - 1}/(h(x - 1))(x + h - 1)/(x + h - 1)
Multiply numerators and denominators:
f'(x) = lim_(hto0){((x - 1)(x + h - 1))/(x + h - 1) - (x + h - 1)}/(h(x - 1)(x + h - 1))
I shall mark what cancels:
f'(x) = lim_(hto0){((x - 1)cancel(x + h - 1))/cancel(x + h - 1) - (x + h - 1)}/(h(x - 1)(x + h - 1))
Remove the canceled factors:
f'(x) = lim_(hto0){(x - 1) - (x + h - 1)}/(h(x - 1)(x + h - 1))
Distribute the -1 in the numerator:
f'(x) = lim_(hto0){x - 1 - x - h + 1}/(h(x - 1)(x + h - 1))
Combine like terms in the numerator:
f'(x) = lim_(hto0){-h}/(h(x - 1)(x + h - 1))
-h/h becomes -1:
f'(x) = lim_(hto0){-1}/((x - 1)(x + h - 1))
It is safe to let hto0:
f'(x) = {-1}/((x - 1)(x - 1))
Simplify:
f'(x) = -1/(x - 1)^2
