Open Intervals Calculus Question

I have to determine the open intervals on which the function f(x)=(x/(x^2+1)) is increasing.
The derivative of this function is (1-x^2)/((x^2+1)^2).
So, where do I need to head from here? Thinking I set the derivative equal to 0, and solve for x's. Those x's are the roots of the equation (maxes/mins), so I can pick points between them and plug into the original formula and see if the line between the two maxima/minima is increasing or decreasing. Is this the right way to approach this?

You are on the right track, you set your derivative to zero to find the critical values. Because the denominator would leave you with non-real answers, only worry about the numerator.

(1-x^2)=0
x^2=1
x= -1,1

Also, note that in the original function, x=0 would make f(x)=0, so on the sign chart, the values for the intervals will be -1, 0, and 1.

f ' (x)______-1______0______1______
(x)

Next, we need to plug in a number from each of those intervals into the derivative function to find whether the answer is positive or negative. I will choose -5, -0.5, 0.5, and 5.

I get the values

f '(-5) = -0.35
f '(-.5)= 0.48
f '(.5) = 0.48
f '(5) = -0.35

f ' (x)___-___-1___+___0___+___1___-___
(x)


From those values we can conclude that the graph is increasing on the intervals (-1,0)U(0,1), or simply (-1,1). Hope that helped.