Given the function f(x)=x^3-3x+3arctan x, prove that f increases if x is real?

The function given is f(x) = x^3 - 3x + 3* arc tan x. If a
function is always increasing its first derivative should always be positive. The first
derivative if f(x)

f'(x) = [x^3 - 3x + 3* arc tan
x]'

=> 3x^2 - 3 + 3/(1 +
x^2)

=> 3*(x^2 - 1 + 1/(1 +
x^2))

=> 3*(x^2 - 1)(x^2 + 1) + 1)/(x^2 +
1)

=> 3*(x^4 - 1+ 1)/(x^2 +
1)

=> 3*x^4 /(x^2 +
1)

Now, if x is real x^2 and x^4 are positive. In 3*x^4/(1
+ x^2) all the terms are positive for real values of x. This verifies that f'(x) is
positive for all real values of x.

As the
first derivative of f(x) is positive for all real values of x, f(x) is an increasing
function.