Determine the second derivative of y=arctanx, if x is in the positive real set.

For the beginning, we'll calculate the first derivative of
f(x):

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

Now,
we'll calculate f"(x) of the expression arctan x, or we'll calculate the first
derivative of f'(x).

f"(x) =
[f'(x)]'

Since f'(x) is a ratio, we'll apply the quotient
rule:

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

We'll put 1' = 0 and
we'll remove the brackets:

f"(x)=
-2x/(1+x^2)^2

Because of the fact that denominator is
always positive, then the numerator will influence the
ratio.

We notice that because of the fact that numerator is
negative over the interval [0,infinite) =>
f"(x)<0.

Therefore, the 2nd derivative
of the function y=arctan x is y"= -2x/(1+x^2)^2 and the graph of the function is concave
over the positive real set of numbers.