A growing square reveals why its area changes at twice its side length.
For the square, (x+h)²−x² = 2xh+h², so dividing by h≠0 gives 2x+h. Letting h→0 yields 2x. In general, the binomial theorem gives (x+h)^n−x^n = n x^(n−1)h plus terms containing h² or higher powers. After division by h, all those remaining terms tend to zero. This proves the rule for positive integers n and all real x.
The square picture covers positive lengths and a positive increment. The binomial argument supplies the general two-sided limit.
The geometric argument uses x > 0 and h > 0 and illustrates the square case. The binomial expansion proves the power rule for every positive integer exponent and all real x.