The Product Rule (Calculus)

Assume u = f(x) and v = g(x) are both positive differentiable functions.  Then the product uv can be interpreted as the area of a rectangle. If x changes by an amount △x, then the corresponding changes in u and v are 

△u = f(x + △x) - f(x) and △v = g(x + △x) - g(x)

And the value of the product (u + △u)(v + △v) can be interpreted as the area of the largest rectangle above. The change in the rectangle then is 

△(uv) = (u + △u)(v + △v) - uv = u△v + v△u + △u△v

Or in other words the sum of the three shaded regions above. Now dividing by △x, yields

△(uv)/△x = u(△v/△x) + v(△u/△x) + △u(△v/△x)

Now take the limit of △(uv)/△x as △x ⟶ 0 and you get the derivative of uv:

d(uv)/dx = u lim△x⟶0(△v/△x) + v lim△x⟶0(△u/△x) +  lim△x⟶0(△u) lim△x⟶0(△v/△x)

d(uv)/dx = u(dv/dx) + v (du/dx) +  0 (dv/dx)

d(uv)/dx = u(dv/dx) + v (du/dx)