More Chain Rule (NancyPi)

Hey, it's Nancy. So in my first chain rule video, I mentioned some problems at the end and then just didn't explain them, left you hanging. Very rude. But a lot of you have asked me to explain them, so I'm going to show you how to use the chain rule to find the derivative if you're given an exponential form like this, a natural log, square root. And also, a bunch of you have asked me how to use the chain rule with the Product Rule. Do you use the chain rule first or the Product Rule first? It can be confusing. Don't worry. I'm going to show you how. Okay. So say you need to find the derivative of something like this. We have an outside function, the exponential e raised to a power, and we have an inside function, 3x, something that's more complicated than just x, so we'll need the chain rule. And if you don't know what I'm talking about, you're going to want to check out my actual intro chain rule video, my first chain rule video. Yeah, that one. And this will all make a lot more sense. But for the chain rule, here's a reminder. First, we take the derivative of just the outside function. So here the outside function is the exponential function e raised to a power. The derivative of the exponential function is just the exponential function. So for the derivative of e, we write e. And when we take the derivative of the outside parts, remember we leave the inside alone. Leave the inside part the same. Just leave it alone like so. We're not done. Remember, we also have to multiply by the derivative of the inside function, the derivative of what's inside the larger outer function. So the derivative of our 3x inside is just three. And now you're actually done. Clean it up a little. It's proper form to bring that coefficient 3 out in front. But that's it. That's your answer for the derivative. dy/dx is equal to 3e^(3x). If this went too fast for you in any way, if you still have lingering questions about this method, doubts, for those of you who are British, please help yourself by checking out my "Chain Rule: How? When? " video, my intro video. So another kind that I mentioned, a natural log example. 5x is what's inside our natural log and the outside function is the ln. So we have ln of something, and that something is more than just x, more complicated than just x, so we use the chain rule. Same steps as before. First, take the derivative of the outside function, of the ln. To take the derivative of ln of something, we write one over that something, one over the inside of the ln. You might be thinking about ln x and how you know its derivative is one over x. But here, ln 5x, because of the chain rule, we write not one over x, but one over the whole inside you actually see here 5x. One over 5x. So we're not done. Remember, we also have to multiply by the derivative of the inside function, the derivative of the 5x inside. Derivative of that is five. So the five and the one-fifth cancel. It simplifies to one over x. Wait. What? Plot twist. It turned out to be one over x anyways. Yeah, that's awkward. It turns out the natural log of a number times x will still have a derivative of just one over x, like ln x does. It's a fun party trick. But the point is that you used the chain rule and you figured that out yourself because you're learning the chain rule. And this could have turned out differently if you had more x terms inside your natural log or powers of x. This wouldn't have simplified to be just one over x. So I just wanted you to be able to handle whatever ln function you're given. To find the derivative, it's going to be one over the inside times the derivative of that inside function. Okay, so let's do this one. Find the derivative, also known as y prime. I gave you Lagrange notation this time instead of the dy/dx Leibniz notation, for those of you who feel strongly about your notation.

Personally, I really prefer this y prime Lagrange notation. Whatever floats your boat. So for this one, we have x^2 plus one under a root, a square root. That square root is our outside, outer function. Whenever you have a root of any kind, before you take the derivative, it's going to be easier to rewrite it in the form of a fraction power. One-half power, so we can use the Power Rule. But our outside function is a power to the one-half. Our inside function is x^2 plus one, something more than just x, so we need the chain rule. To take the derivative of the outer one-half power, we'll use the Power Rule and bring down the one-half power to the front like that. And leave the inside function the same here. Leave the inside alone. We brought the power one-half down to the front. What's the new power going to be? By the Power Rule, we reduce the old power by one. So one-half minus one is negative one-half. Okay. And don't forget. You're not done. We have to multiply by the inside derivative. We just focus on our inside x expression. The derivative of x^2 plus one is 2x plus nothing, just 2x. And you're done. So the one-half canceled with the two. It's considered nicer, proper, to put it back into a root form A lot of people won't like you leaving a negative power in your final answer. This negative one-half power means one over the one-half power or one over square root. That's why we've written here in the final answer, one over root form. Why did we bother even putting it into a power form if we're going to put it right back into a root form? Because we're masochists. No. It was just easier to use the Power Rule in the power form. Okay, so a bunch of you have asked me, "What do you do if you have the chain rule combined with the Product Rule? Do you use the chain rule first? Or not? " Good question. It's not obvious. By the way, if you don't know the Product Rule, you can check out my "Derivatives: How? " video to see that. But let's look at our equation here. If you had only this half, 2x minus five to the fourth power, and not this, you could definitely use the chain rule on that to find the derivative. We have this outer function to the power of four, inside function that's 2x minus five. I know I sound like a broken record, but our inside is more than just x, more complex than just x, so we could use the chain rule on that. It would be a lot like number one in my first chain rule video, the very first one I showed you in the main video. But we don't have just that. It's also multiplied by this x^3. So in the larger function, the big picture view, the biggest view, what we have is two things multiplied together, the product of two things. So that's your answer. You're going to use the Product Rule here first because in the largest view, we have a product of two things, first and foremost. Just to remind you, the gist of the Product Rule is that when you have two functions multiplied together, the derivative will be the first function, as is, times the derivative of the second, plus the second function, as it is, times the derivative of the first. So let's try that for our equation. So we write the first factor as is, x^3, times the derivative of the second factor. Now, this is the moment where we use the chain rule within the Product Rule because what we're about to take the derivative of is an outside function with an inside more than just x. So for the derivative of the outer function, the power of four, by the Power Rule, we bring down that power to the front and reduce the old power by one. Multiply it by the derivative of the inside function. Derivative of 2x minus five is just two. For a fuller, slower explanation of using the chain rule on a power like this, see that example number one in my first video. We're only halfway done. We're going to add the second function times the derivative of the first. Plus The second function times the derivative of x^3 is just 3x^2.

And that's it. That's the derivative. We can clean it up, simplify. If we want to, or if we have to. You can clean up the coefficients in each of the terms. And then it's considered nicer and proper to pull out common factors to the front. So there's an x^2 that appears in both of these terms, and there's a 2x minus five to the third power that's contained in both of these terms. You can pull those out to the front. I know it feels like grunt work sometimes to simplify this way, but we get this pretty little result in the end with three factors just multiplied together, so it cleans up nicely. So this is how you do the chain rule with the Product Rule. Chain rule inside the Product Rule. Will it always be Product Rule first? I'm going to be honest. That is usually what you'll see, but chain rule first with the Product Rule can also happen. Like what if our product were inside a natural log? Then you'd use the chain rule first on the natural log. And when you go to take the inside derivative, since the inside is a product, you'd use the Product Rule then and not first. It's less common to see, but basically the first thing you'll do is whatever you need to do to handle the outermost form. So I hope that helped you understand more ways you'll need the chain rule. I know calculus is everyone's favorite. You don't have to like math, but you can like my video. So if you did, please click Like or Subscribe.