For x>0, find and simplify the derivative of f(x)=arctanx+arctan(1/x).

Answer 1

Using the fact that:

the first half is trivial. The second half requires us to use the chain rule so:

the derivative of arctan(1/x) = 1/(1+(1/x)²)•1/x)' = 1/(1+1/x²)•(-1/x²) = -1/(x²/x² + 1/x²)•(1/x²) =

-1((x²+1)/x²)•(1/x²) = -x²/(x²+1)•(1/x²) = -1/(x²+1)

So f '(x) = 1/(1+x²) - 1/(x²+1) = 0