3-B Product Rule
Instructions
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.
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 . So, to visualize a change in the product function, we have to investigate what a change in the area looks like.
3-B The Product Rule
The Product Rule (for differentiation) states how the differentiation operator interacts with the function operation of multiplication. That is,
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 and , we can express the rule a little more neatly:
An important observation with this rule is that the differentiation operator does not "distribute" across multiplication. The derivative of a product is NOT the product of the derivatives.