To find the difference quotient at any point (x,f(x)), we need to find f(x) for a value of x just a tiny bit bigger than x. This value of x has a value of x+h, where h is a very tiny quantity. f(x+h)=(x+h)^3. This can be expanded as x^3+3x^2h+3xh^2+h^3. The difference between this and f(x)=x^3 is therefore 3x^2h+3xh^2+h^3. The difference in the x value is, of course, h by definition of h. So the difference quotient is (3x^2h+3xh^2+h^3)/h, being the vertical displacement divided by the horizontal displacement. This comes to 3x^2+3xh+h^2. Because h is very tiny, infinitesimally small, in fact, we can ignore the terms involving h, leaving us with 3x^2, the difference quotient. The difference quotient can be used to calculate approximations in evaluating the cubes of numbers close to known cubes. If we want the cube of 3.05, we know that 27 is the cube of 3, so we add 3*9*0.05=1.35 to 27 to get an approximation, i.e., 28.35. The actual value is closer to 28.373.