1/x and y intercepts

2/co-ordinates of all critical values

3/classify the critical values using the second derivative test

4/determine the coordinates of all possible points of inflection

5/check to make sure there is a change in concavity at these points

6/determine all increasing and decreasing intervals

7/determine the intervals of concavity

8/determine the equation of the horizontal asymptote

9/provide a sketch and label all parts of the graphical analysis determined above

so for f'(x) i get \(\displaystyle f'(x)=\frac{2(x^2-1)}{(x^2+1)^2}\) and for f''(x) i get \(\displaystyle f''(x)=\frac{-4x(x^2-3)}{(x^2+1)^3}\) both of which i got using the quotient rule, although very unfamiliar as to how to work out each step