Product Rule f(x) = x(2x-1)(2x+1) Hi! Welcome back. Another product rule problem, this time with three terms. So, f(x)=x(2x-1)(2x+1). So, the primary point of this is just to demonstrate how to apply product rule when you’ve got three terms. The first term is x, the second term is 2x-1 and the third term is 2x+1. So, same as before, you take the derivative of one term and you keep the other two constant then you take the derivative of another term, keep the other two constant and so on. You could do this with an infinite number of terms. So, the derivative we’re going to go ahead and say f1(x) = and we’re going to go in order. You take the derivative of the first term x, the derivative of x is just 1, so that’s implied. We keep the other two the same, so (2x–1)(2x+1). Then adding, always adding, so plus. We’re going to go ahead and keep this one the same x, take the derivative of the second one. The derivative of 2x–1 is just 2, so we can go ahead and write that in front, 2x; 2 comes from here, x is the same right here and then we multiply by 2x+1 because this remains constant. Then finally, we’re going to add the third term. We’re taking the derivative of the third one this time keeping these two the same. The derivative of this term is just 2, so we’re going to go ahead and move that out with plus 2 and then we keep this 2. There’s the x and then times 2x-1. So, that is fundamentally what product rule is. If you were asked this on a test and you would had to find the derivative and it wasn’t necessarily demonstrate product rule, you could extend all of these, multiply everything out and then take the derivative term by term. If you’re asked to show product rule, you’re going to have to do it this way and then I’m guessing that almost certainly, you’re going to have to simplify this. Let’s just go ahead and do that really quickly. 2x×2x=4x2, 2x×1 is going to be plus 2x, this two, -2x and then -1×1 is -1 and then plus. Now, we’re on to this part and I’m going to go ahead and multiply that out 2x×2x = 4x2, 2x×1+2x and then plus, I’m going to go ahead and multiply this out as well. 2x×2x is 4x2 and 2x×-1-2x, so then we can simplify. We’ve got 12x2 and then 2x-2x is zero plus 2x is 2x-2x is zero and then -1 and that’s everything. So, 12x2–1 and that is our final answer as the derivative of this problem showing you how to apply product rule to this function. Thanks.