Guessing functions given initial conditions

[x3ro]

Is there any way to mathematically derive functions that satisfy a given set of initial conditions? I know this sounds very general, but say for:

f(1)-f(0)=0

and

f'(1)=1

I've resolved myself to guessing and checking. I've found a function for the opposite: e*t-exp(t) -- ([e*1 - exp(1)] - [e*0 - exp(0)] = [0]-[-1]= 1 -- derivative is e - exp(t) --> (e - exp(1) = 0) but given how long it took just to guess that function, I was hoping for a more concrete way to find these functions.

 

[x3ro]

Of course after posting this, I come up with a very simplistic solution...x^2 - x. But, had to guess...so still doesn't solve the problem for me.