Determine f'(1) , f(x)=(x+5)/(2x+1).

To calculate f'(1), we'll use the definition of the
derivative:

f'(1) = lim {[f(x) - f(1)]/(x-1)} ,
x-->1

f(1) = (1+5)/(2+1) = 6/3 =
2

f'(1) = lim {[(x+5)/(2x+1)  - 2]/
(x-1)},   x-->1

f'(1) = lim (x + 5 - 4x - 2) /
(x-1)(2x+1), x-->1

We'll combine like terms from
numerator:

f'(1) = lim (-3x + 3)/
(x-1)(2x+1), x-->1

We'll factorize by
-3:

f'(1) = lim -3(x - 1)/ (x -
1)(2x+1),  x-->1

We'll reduce like
terms:

f'(1) = lim -3 /
(2x+1),  x-->1

We'll substitute x by
1:

f'(1) =  -3 / (2+1)

f'(1)
= -3/3

f'(1) =
-1

We could also use the rule of
quotient;

(u/v)' = (u'*v -
u*v')/v^2

u = (x+5) => u' =
1

v = (2x+1) => v' =
2

(u/v)' = [(2x+1) -
2*(x+5)]/(2x+1)^2

We'll remove the
brackets:

(u/v)' = (2x+1 -
2x-10)/(2x+1)^2

We'll eliminate like
terms:

f'(x) = (u/v)' =
-9/(2x+1)^2

We'll calculate
f'(1):

f'(1)
=  -9/(2*1+1)^2

f'(1)
=  -9/3^2

f'(1)
=  -9/9

f'(1)
=  -1