Why is the golden ratio lurking in $(d/dx)\arctan\left( x + \frac{1}{x} \right)$ $$ = \frac{\left(\frac{1+\sqrt{5}}{2}\right)}{x^2 + \left(\frac{1+\sqrt{5}}{2}\right)^2} + \frac{\left(\frac{1-\sqrt{5}}{2}\right)}{x^2 + \left(\frac{1-\sqrt{5}}{2}\right)^2}\text{ ?} $$ Is this merely an instance of its (unbeknownst to me) lurking everywhere, or is something special about this particular arctangent of a sum?

(An arctangent of a sum seems like a bit of a freak, though.)

(This was inspired by a related question that someone posted to http://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Mathematics  .)

Why is the golden ratio lurking in $(d/dx)\arctan\left( x + \frac{1}{x} \right)$ $$ = \frac{\left(\frac{1+\sqrt{5}}{2}\right)}{x^2 + \left(\frac{1+\sqrt{5}}{2}\right)^2} + \frac{\left(\frac{1-\sqrt{5}}{2}\right)}{x^2 + \left(\frac{1-\sqrt{5}}{2}\right)^2}\text{ ?} $$ Is this merely an instance of its (unbeknownst to me) lurking everywhere, or is something special about this particular arctangent of a sum?

(An arctangent of a sum seems like a bit of a freak, though.)

(This was inspired by a related question that someone posted to http://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Mathematics.)

Why is the golden ratio lurking in $(d/dx)\arctan\left( x + \frac{1}{x} \right)$ $$ = \frac{\left(\frac{1+\sqrt{5}}{2}\right)}{x^2 + \left(\frac{1+\sqrt{5}}{2}\right)^2} + \frac{\left(\frac{1-\sqrt{5}}{2}\right)}{x^2 + \left(\frac{1-\sqrt{5}}{2}\right)^2} $$ Is this merely an instance of its (unbeknownst to me) lurking everywhere, or is something special about this particular arctangent of a sum?

(An arctangent of a sum seems like a bit of a freak, though.)

(This was inspired by a related question that someone posted to http://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Mathematics .)

Why is the golden ratio lurking in $(d/dx)\arctan\left( x + \frac{1}{x} \right)$ $$ = \frac{\left(\frac{1+\sqrt{5}}{2}\right)}{x^2 + \left(\frac{1+\sqrt{5}}{2}\right)^2} + \frac{\left(\frac{1-\sqrt{5}}{2}\right)}{x^2 + \left(\frac{1-\sqrt{5}}{2}\right)^2}\text{ ?} $$ Is this merely an instance of its (unbeknownst to me) lurking everywhere, or is something special about this particular arctangent of a sum?

(An arctangent of a sum seems like a bit of a freak, though.)

(This was inspired by a related question that someone posted to http://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Mathematics .)