prove that d/dx arctan x = 1/(1+x^2)

the solution given by the lecturer...

(f-1)'(x) = 1/f'(f-1(x)) ....... by theorem 1

= 1/cos(arcsin x)

= 1/sqrt(1-sin^2(arcsin x)

= 1/sqrt(1-x^2)

just wandering how theorem 1 is derived.... =p anyone can explain to me... need to understand why rather than memorizing the formula...