The graphs of two functions are shown on the left. Two points and the secant line through them are displayed. These can be adjusted from the slider tools for c and h on the right. [br][br]The visual on the right is a rectangle, the sides of which represent the values of the two functions, f and g. In other words, the area of the rectangle represents the product [math]f(c)\cdot g(c)[/math]. So, to visualize a change in the product function, we have to investigate what a change in the area looks like.

The [b]Product Rule[/b] (for differentiation) states how the [i]differentiation operator[/i] [math]\frac{d}{dx}[/math] interacts with the function operation of [i]multiplication[/i]. That is,[br][br][math]\frac{d}{dx}\left[f(x)\cdot g(x)\right]=g(x)f'(x)+f(x)g'(x)[/math][br][br]This function notation is nice because it reminds us that we are dealing with functions (not numbers), but it is also a little cumbersome. If we let [math]u=f(x)[/math] and [math]v=g(x)[/math], we can express the rule a little more neatly:[br][br][math]\frac{d}{dx}\left[uv\right]=vu'+uv'[/math][br][br]An important observation with this rule is that the differentiation operator does not "distribute" across multiplication. [i]The derivative of a product is NOT the product of the derivatives[/i].