That formula makes sense when you draw a velocity vs time graph and realise the area under the curve is the distance travelled. With linear acceleration, the area is a rectangle with a triangle on top, and everything starts becoming clear, even without calculus.
You can also visualize the meaning of d/dx x^n = n x^(n-1) geometrically: There are n faces being extruded which contribute a volume of x^(n-1) dx each.
