Will you please find all values of x for the given function where the tangent line is horizontal?
f(x) = x / (x^2 + 1)^3
The slope of a line that is tangent to a certain point on f(x) is different at different points because f(x) increases and decreases at different rates. Taking f'(a) will tell you the slope of a line that is tangent to the point (a, f(a)) on the graph of f(x), but the slope of the point (a+1) is going to be different.
You are right though that you need to find when f'(x)=0 and solve for x.