To find the difference quotient consider f(x+h), where h is a very tiny value. If we evaluate this we get f(x+h)=(x+h)^2+5, which expands to x^2+2xh+h^2+5. If we subtract f(x)=x^2+5 from this we get 2xh+h^2. The difference quotient is this divided by h, giving us 2x+h. We ignore h because it's very tiny, infinitesimal. So the difference quotient is 2x. Note that the constant 5 disappears and the power of x is reduced.