Any function, no matter how crazy or nonlinear, looks like a straight line if you zoom in far enough. When you're looking at the value of a function at two points separated by an infinitesimal distance, dx, that's the ultimate manifestation of that same effect. As dx approaches zero, the function (ANY function!) looks like a straight line over that interval, and it's slope is df(x)/dx. Hence, you can approximate the function over that interval with the equation of a line. If we use the form (yy0)=slope*(xx0), we can substitute your values:
f(x+dx)f(x)=d/dx f(x)*((x+dx)x),
which simplifies to
f(x+dx) = f(x) + d/dx f(x) * dx.
