Differentiate the function by forming difference quotient [f(x+h)-f(x)]/h and taking the limit as h tends to 0. f(x)=Square Root(x-1)

To calculate the value of the first derivative in a given
point, h = 0, we'll have to apply the limit of the
ratio:

limit [f(x+h)-f(x)]/h, when h tends to
0.

We'll calculate the value of f(x+h) and we'll substitute
f(x+h):

f(x+h) = f(x+0) =
f(x)

The limit will
become:

limit [f(x+h)-f(x)]/h = limit [f(x)-f(x)]/h , h
-> 0

limit [f(x)-f(x)]/h = 0/0
indetermination case

We'll apply L'Hospital
rule. We'll differentiate separately both numerator and denominator, with respect to
x.

[f(x)-f(x)]' = f'(x) -
f'(x)

f'(x) =
[sqrt(x-1)]'

f'(x) =
(x-1)'/ 2sqrt(x-1)

f'(x) =
1/2sqrt(x-1)

 f'(x) - f'(x) = 1/2sqrt(x-1) - 1/2sqrt(x-1) =
0

We'll differentiate the denominator h(x) =
(x-0)

h'(x) = x'

h'(x) =
1

limit [f(x)-f(x)]'/h'(x) =
0/1

limit [f(x+h)-f(x)]'/h'(x) =
0