 What if we want to find the derivative of a product? Well, we could use something called the product rule, but let's make sure we have some assurance our answers are correct. For example, let's find the derivative of this product, and the idea here is that we want to know what the derivative actually is before we apply some new fangled rule to it. So let's do some algebra, and that gives us a nice polynomial we can differentiate. So expanding, then differentiating, gives us, and you should view this as an answer we trust. So maybe we can find the derivative of a product as the product of the derivatives. Well, let's find out. The derivative of this product, maybe it's the derivative of 2x minus 7 times the derivative of x squared plus 3x minus 5. Well, we can differentiate, and we get a possible derivative of 4x plus 6, but when we expand it and differentiate it, we found and trusted the answer, 6x squared minus 2x minus 11, and so this rule doesn't work. It can't be correct. So let's go back to basics. According to the definition, the derivative is the limit of the difference quotient. At this point, we have to do a lot of algebra to make this work out, and the thing we need to do is far from obvious. We're going to subtract and add the same quantity, f of x plus h times g of x. Now we can split apart this rather complicated rational expression, and remember the limit of a sum is the sum of the limits. So let's take a look at the two individual limits. Now this first limit can be simplified a little bit. We can factor out this common factor of f of x plus h. The limit of a product is the product of the limits. Now this first factor, as x goes to h, that just goes to f of x. But the second factor has a very specific form, and it is in fact the definition of g prime of x. How about that second limit? This time everything has a factor of g of x, so we can remove that factor. The limit of a product is the product of the limits, and this limit is just the derivative of f. So the second term is g of x, f prime of x, and so the derivative of a product is the sum of the products of one function times the derivative of the other. This gives us the product rule for derivatives. The derivative of a product, f of x times g of x. Well, that's f g prime plus g f prime. So let's go back to that polynomial product whose derivative we actually know if we apply our product rule. We want the first factor times the derivative of the second, plus the second factor times the derivative of the first, and these are both derivatives of polynomials, so we can find those directly, and we get our final answer, and if we want to check, we can verify that this is equal to 6x squared minus 2x minus 11, just as we hoped.