Ok! just use the chain rule in doing the derivative. i.e. if y=f(g(x)) ==> y'=g'(x)*f'(g(x)). Now see how this rule applies to the question.
let g(x)=1-x, now [f(1-x)]'=-f'(x). Finally set x=0 and obtain. f'(0) [comes from the LHS of equation] = -f'(1) [which comes from the RHS of the equation].
That's all. :d