 Let's recap the main ideas and results of section 2.1 and active calculus on elementary derivative rules. So the main idea of this section, and indeed this whole chapter, is to use the definition of the derivative as we developed it in chapter one. And try to come up with ways of calculating derivative formulas quickly by exploiting the patterns that we see when we use that limit definition. In this section we encounter several fundamental computational rules for derivatives and it's imperative that you master these basic rules as early as possible. First of all, we saw that the derivative of any constant function, that is a function whose output is always the same value, is equal to zero. This makes sense on a number of levels since, for example, if the derivative of f tells us the rate at which f is changing, then if f is never changing, we'd expect its derivative to be zero. And indeed it is. Next, we learn that if f is a power function, that is, its formula is given as f of x equals x to the nth power, where n is a non-zero constant power. Then the derivative of f is f prime of x equals n times x to the n minus 1 power. This formula works for any non-zero power of x, including negative powers and fractional powers. So functions like the square root of x, which is x to the 1 half, and 1 over x, which is x to the negative 1, are covered by this derivative rule. Very important thing to note here. This rule does not apply to absolutely any function that has an exponent on it. For example, exponential functions such as 2 to the x power are like power functions, except that the roles of the base and the exponent are switched. Whereas a power function has a variable base and a constant exponent. Exponential functions have a constant base and the exponent is variable. So for example, exponential functions do not follow this power rule. A crucial principle in using these fast differentiation methods is to take them very literally and do not use them on a function unless the function fits the rule's description exactly. So the power rule doesn't work on exponential functions. We have a separate rule for those, namely this. That for any positive real number a, if f of x is the exponential function a to the x, then f prime of x is a to the x times the natural logarithm of a. We'll see why this derivative rule is what it is in a few more sections. Lastly, there were two more fundamental rules in this section that work under general conditions. One is called the constant multiple rule that says that for any real number k, if f of x is differentiable, then the derivative of k times f of x is equal to k times the derivative of f of x. Notice two things here. One, we are using the d over dx notation that was introduced in this section. Remember the d over dx followed by a function just means the derivative of that function. And two, in this rule, k is a constant. If k is a variable, then this rule does not apply. The other general rule is called the sum rule, which says that if f of x and g of x are differentiable functions, then the derivative of f of x plus g of x equals the derivative of f of x plus the derivative of g of x. The main goal for right now is to learn to use these rules, both individually and in combination with each other, with absolute fluency. So that you are automatic when you're computing with them by hand. That's the thrust of the activities in the section. And we'll now see some examples of those rules at work in the remaining videos to give you a start.