Applications of the first derivative?

[cmnalo]

I need to find the interval where the function is increasing or decreasing?

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

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

f'(x)= (x+2)^2 (2) - 4x^2 +4x / (x+2)^4

I'm stuck on the next step? I think I need to simplfy further and than I need to find the interval. Any help would be appreciated.

 

[skeeter]

I need to find the interval where the function is increasing or decreasing?

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

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

f'(x) = [(x+2)²(2) - (2x)2(x+2)]/(x+2)⁴
factor out the common factors from the two terms in the numerator ...
f'(x) = 2(x+2)[(x+2) - (2x)]/(x+2)⁴
f'(x) = 2(2-x)/(x+2)³

critical values are x = 2 and x = -2 ... check the sign of f'(x) in the three intervals of the real numbers formed by the two critical values to see where f(x) is increasing and decreasing.


f'(x)= (x+2)^2 (2) - 4x^2 +4x / (x+2)^4

I'm stuck on the next step? I think I need to simplfy further and than I need to find the interval. Any help would be appreciated.

 

[cmnalo]

I'm having trouble following how you reduced the numerator. I don't understand how you got from this:
(x+2)^2 (2) - (2x) [2(x+2) (1)] / (x+2)4


To this:
f'(x) = 2(x+2)[(x+2) - (2x)] / (x+2)4

 

[skeeter]

I'm having trouble following how you reduced the numerator. I don't understand how you got from this:
(x+2)^2 (2) - (2x) [2(x+2) (1)] / (x+2)^4

note the two terms in the numerator ...

2(x+2)² and 2(2x)(x+2)

what factors do each term have in common?

To this:
f'(x) = 2(x+2)[(x+2) - (2x)] / (x+2)^4