Differentiate the following w.r.t.x. (x+1)/(x+2)^2 (3x+4)/(4x+5)

We'll differentiate the first ratio, with respect to x,
applying the quotient rule:

(f/g)' = (f'*g -
f*g')/g^2

(x+1)/(x+2)^2 =
f(x)/g(x)

We can write the differentiating process in 2
ways:

d/dx[(x+1)/(x+2)^2] = [(x+2)^2d/dx(x+1) -
(x+1)d/dx)/(x+2)^2]/)/[(x+2)^2]^2

or

[(x+1)/(x+2)^2]'
= {(x+1)'*(x+2)^2 -
(x+1)*[(x+2)^2]'}/[(x+2)^2]^2

[(x+1)/(x+2)^2]' = [1*(x+2)^2
- 2(x+1)(x+2)*(x+2)']/(x+2)^4

[(x+1)/(x+2)^2]' = [(x+2)^2 -
2(x+1)(x+2)]/(x+2)^4

We'll factorize the numerator by
(x+2):

[(x+1)/(x+2)^2]' =
(x+2)*(x+2-2x-2)/(x+2)^4

We'll combine like terms and we'll
simplify:

[(x+1)/(x+2)^2]' =
-x/(x+2)^3

d/dx[(x+1)/(x+2)^2] =
-x/(x+2)^3

Now, we'll
differentiate the ratio (3x+4)/(4x+5) using again the quotient
rule:

d/dx[(3x+4)/(4x+5)] =
[(3x+4)'*(4x+5)-(3x+4)*(4x+5)']/(4x+5)^2

d/dx[(3x+4)/(4x+5)]
= [3(4x+5) - 4(3x+4)]/(4x+5)^2

We'll remove the
brackets:

d/dx[(3x+4)/(4x+5)]
=(12x+15-12x-16)/(4x+5)^2

We'll eliminate like
terms:

d/dx[(3x+4)/(4x+5)] =
-1/(4x+5)^2