In this problem, we're asked to find the derivative of two different functions. Our first one is f(x) = sin5(x), and our second function is sin(x5). Now we want to find both of these derivatives, which we can do using the chain rule. So let's look at our first function here, sin5(x). Now, this might not be immediately clear how we're going to use the chain rule here, just because of how this is written.
But remember that if I have the function sin2(x), this is equivalent to saying sin(x), that whole function, squared. We just usually write it with that power, that exponent, in the middle of the sin and the x. So here, when I have sin5(x), I can also think of this as being sin(x) all to the power of 5. This makes it much more clear what our inside and outside functions are so that we can use the chain rule. So let's go ahead and get started here using our chain rule to find f'(x), our derivative.
Now here I have 2 functions. Of course, I have my inside function, which is sin(x), and I have my outside function, which is all of this stuff raised to the power of 5. Now, even though this has a trig function, our chain rule still works in the same exact way. We're going to start from the outside and work our way inside. So starting with that outside function here, we have all the stuff raised to the power of 5.
We're going to pull that 5 out to the front using the power rule and decrease that power by 1. So this becomes 5 times the sin(x), that's what my inside function is, all raised to the power of 4. Now working our way inside, we want actually to take the derivative of that inside function sin(x) and multiply by it.
Now, what is the derivative of sin(x)? We know that it is cos(x). So this is all multiplied by cos(x). Now we can rewrite this slightly because we know that the sin4(x), we're going to pull that 4 back inside just to write it by convention. So we have that 5 on the outside.
I'm actually going to write my cos(x) first here since it's not raised to a power. I have 5 cos(x) times sin4(x). And this is my final derivative, 5 cos(x) times sin4(x). Now that is our first function, we found the derivative here. Let's move on to our second function.
Now our second function here is still going to be using the chain rule. We have sin(x5). So here, it's a little bit more clear initially that I have my inside function x5, and I have my outside function, sin. So let's go ahead and see what this looks like when we apply the chain rule to find f'(x). Now, even though we have a trig function on the outside, we are still starting from the outside working our way in.
So we want to start by finding the derivative of that outside function. Now here, our outside function is sin. What is the derivative of sin? It is cos. So here, I have cos.
What am I taking the cos of? Well, since we're using the chain rule here, remember that we're keeping our inside function the same. So here, I'm taking the cos of x5, keeping that argument as is because that is my inside function. But we are not done yet because using the chain rule, we also need to take the derivative of that inside function and multiply by it. So I have cos(x5).
I am multiplying that by the derivative of x5. Using the power rule, moving that 5 to the front, decreasing that power by 1, that gives me 5x4. And this is my derivative. Now, conventionally, we typically want to write that variable and the constant upfront, 5x4 times cos(x5). So this is my final answer for my derivative here that I also found using the chain rule.
Now, it's going to be really important here to keep getting more practice because you want to get really, really comfortable using the chain rule no matter if you have trig functions or not. Let me know if you have any questions, and I will see you in the next one.
