# Exploring the Product Rule (Activity 12 HL)

- Author:
- Callum Marshall, Jon Dreyer, syjang

- Topic:
- Differential Calculus

This is a visualization of the product rule in calculus for functions that are the product of two functions.
For example, . This is a product of and ;
We are going to look at a function, The basic idea is that

**the product of two functions can be visualized as the area of a rectangle.****NOTE:**the curve below is**PARAMETRIC**, meaning that each point on the curve represents (u(t), v(t)) for some t, NOT the point (t, f(t)).## QUESTION 1

In the applet

**ABOVE****SET h = 0;**we will define the rectangle you see as our INITIAL rectangle.

**SET h = 0.6;**calculate the INCREASE in the area by calculating the sum of the areas of the blue, green and brown rectangles. Then, divide this area by the value of*h*, giving your answer correct to 3-significant figures.

**SET h = 0.4;**repeat what you did above.

**SET h = 0.2;**repeat what you did above.

**SET h = 0.1;**repeat what you did above. What do you notice about the values you calculated as you reduced the value of h?

## QUESTION 2

Consider the expression shown in the applet

**BELOW**. What does**NUMERATOR**represent?**Set the value of**As you move the value of*t*to*t*= 2.*h*, closer to 0 what does the value of the fraction represent?**Set the value of**Again as the value of*t*to*t*= 1.*h*moves closer to 0 what happens to the value of the fraction? What do the values you have found represent?## QUESTION 3

The
Show how to use the expression in the applet ABOVE to derive this rule.

**PRODUCT RULE**says that;**HINT 1:**split the fraction up into THREE separate fractions...then go from there!**HINT 2:**Recall that;