oops I meant "Rate of change of area of a square with respect to its side length" Ok I have to use this annoying Stewart textbook for my Calc class in college. Most of the questions require what I like to call "Monkey Math," where you just memorize a set of steps and then follow them rigidly for each and every problem. However, this problem I found has me really thinking, here it is: Show that the rate of change of the area of a square with respect to its side length is half its perimeter. Try to explain geometrically why this is true. So I can easily "show" why this is true. A(x)=x2 A'(x)=2x P(x)=4x so (A'(x))/P(x)=1/2 But I am drawing this out on my white board and I cant conceptually understand why the rate of change of the area would be 1/2 the perimeter at the exact same moment... I can understand why this is true algebraically, but I guess I can't visualize what the derivative of a geometric shape's area is. Can anyone shed some light on this for me?