Given that, f(x-y)=f(x)/f(y).......(star).
x=y, &, (star) rArr f(0)=f(y)/f(y)=1.......(1).
x=0, y=y, & (star) rArr f(-y)=f(0)/f(y)=1/f(y).........(2).
x=x, y=-y, & (star) rArr f(x+y)=f(x)/f(-y)=f(x)f(y)
:.," by "(2), f(x+y)=f(x)f(y).............(3).
Knowing that, f'(x)=lim_(h to 0) {f(x+h)-f(x)}/h, we have, by (3),
f'(x)=lim_(h to 0){f(x)f(h)-f(x)}/h=f(x){lim_(h to 0) (f(h)-1)/h}...(ast).
Now, f'(0)=p, (ast), & (1) rArr p=lim_(h to 0) (f(h)-1)/h...(4).
Similarly, f'(5)=q, (ast), & (4) rArr q=p*f(5)......(5).
"Finally, by "(ast), &, (4), f'(-5)=p*f(-5)
=p*(1/f(5)),............[because,(2)]
=p/(q/p),................[because, (5)]
:. f'(-5)=p^2/q, as Respected Cesareo R., has derived!
Enjoy Maths.!