 Let's recap the main ideas from section 2.3 of active calculus on the product and quotient rules. In this section, we added two very important techniques to our toolbox for calculating derivatives. The first is called the product rule, and we use it in situations where we're taking the derivative of two functions that are being multiplied together. If f and g are differentiable functions and p is their product, then the derivative of that product, according to the product rule, is f times g prime plus f prime times g. In other words, the derivative of the product of two functions is the first function times the derivative of the second plus the derivative of the first times the second. Importantly, notice that the derivative of f times g is not just f prime times g prime. The second technique from this section is the quotient rule for differentiating two functions that are being divided. If f and g are differentiable functions and q equals f over g is their quotient with g of x not equal to zero, then the derivative of that quotient is the following. f prime times g minus f times g prime divided by g squared. Verbally, the derivative of f divided by g is the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator divided by the denominator squared. An important difference between the quotient rule and the product rule other than the fact that it's a fraction here in the quotient rule is the presence of the minus sign. This should tell you that the order in which we set this expression up here in the numerator matters. Notice that the derivative of f over g is not just f prime over g prime. In the textbook, there is a derivation that shows you how the quotient rule comes about. Basically, we get the quotient rule by just using the product rule. This is an interesting argument that explains why the quotient rule is what it is, and you should take time to read that. In the next few videos, we'll be looking at examples of usage for both the product and the quotient rules.