 Let's find the derivative of the function y equals 4x times e to the x plus five Race of the fifth power. So if we're going to calculate the derivative of this thing We have to of course to take the derivative of 4x times 3x plus five to the fifth power For which you'll notice there's a couple things going on here But the first thing to note here is that this is a product of two different functions. We have 4x times 3x plus five to the fifth power and so before We can do anything else the first thing we have to do here is apply the product rule for which as we know about the product Well, we're gonna take the derivative of 4x and then times that by 3x plus five to the fifth power And then we add to that 4x times the derivative of 3x plus five To the fifth power so we're gonna calculate that derivative the derivative of 4x is simple enough The derivative with respect to x is just gonna be a four so we get four times 3x plus five to the fifth power Copy down the 4x here then the next part is where it might be the trickiest here You have 3x plus five to the fifth power Which derivative rule are we gonna use in this case? It's gonna be the chain rule because you'll notice you have this inner function 3x plus five We also have this outer function. That is the fifth power. It's a power function there So by the chain rule we take the outer derivative first by the power rule The outer derivative is gonna be five times 3x plus five to the fourth power the power goes down by one Then we have to take the derivative of the inner function the derivative of 3x plus five Which the inner derivative there is gonna be a three It's perhaps the most common mistake for students to forget that inner derivative you want to watch out for that thing So what do we have here? We have four times three x plus five to the fifth plus 4x times five times three x plus five to the fourth times three This is the derivative of the function now I should mention in terms of calculus we're now done, but oftentimes we have to solve For the derivative equal to zero which really we want it to be factored then situation So let's see what would happen if we try to factor this thing some things to note is there is a common factor of four But there's also a common factor of three x plus five The first product has one and the second product has one you cannot take more than the least holder here That is since one is the fifth power once the fourth power You can't take away more than what the least product has which is gonna be the fourth power So we're gonna factor out a four. We're gonna factor out a three x plus five to the fourth power Three x plus five to the fourth power What's been left behind with the first product we took away the four We took away four of the three x plus five so at least behind a three x plus five For the second group. Let's see we took away the four. We left behind an x We didn't take the five, but we took away all the three x plus fives. We're left behind with a three So that's then in the brackets is what we need to combine together to try to simplify For the first one we just dropped the parentheses three x plus five for the next one We're gonna end up with a 15 x which is just five times three right there. And so combining together the like terms in those brackets we end up with a 18 x plus five which is our derivative Which we had a derivative earlier, right? I mean the derivative was calculated here No more calculus was necessary at that point like I said It's to our benefit to get in the practice of facting our derivatives So this is the final form that we want to record as our final answer