Let's start by using calculus to prove that the slope of a horizontal line is, in fact, zero. This will establish our first shortcut, the Constant Rule. Duration: 6:54

For Investigation 1, you are trying to discover a general rule for taking the derivative for a power function. Do accomplish this noble task, use the limit definition to find the derivative of y=x, y=x^2, y=x^3, y=x^4, and y=x^5. Hopefully, that will be enough to establish a pattern.

Let's debrief and recap the results you notice from your investigation. If you extend the pattern, what would you get for y=x^6 or y=x^n? Duration: 7:57

BC ONLY. Our goals now is to use the limit definition to prove that the derivative of x^n is nx^(n−1). Before we do, let's refresh our knowledge of Pascal's Triangle. Duration: 7:07

BC ONLY. Before watching the video below, solve this problem: In a group of 5 people, how many pairs of people can you choose, assuming order does not matter? Now let's examine that people-pairing problem in an organized fashion and then show how we could have easily answered that question using the combination formula. By the way, on the TI-84, the combination operation is found under the MATH menu: MATH > PROB > 3: nCr. Duration: 7:52

BC ONLY. Finally, we are ready to prove the Power Rule using the limit definition of the derivative. Fairly straight forward. Duration: 12:29

After formally stating both the Constant Rule and the Power Rule for derivatives, we give them a spin on a handful of examples. What used to take hours, now takes seconds. Duration: 8:18

In ordinary circumstances, we use the derivative to help us write the equation of a tangent line. For this example, we will "recipnegate" the slope and also write the equation of the normal line. Duration: 4:02

The Constant Multiple Rule, our next derivative shortcut, asks what happens to the slope of a function when you vertically stretch or shrink it. Duration: 5:38

What do you think is the derivative of the sum of two functions. Either the Sum or Difference Rule will answer that question. Duration: 4:46

Now that we have a number of derivative rules under our belt, we can easily take the derivative of any polynomial function, as these examples clearly show. Duration: 9:48