In this example problem, we're asked to find the derivative of the function . Now looking at this problem, it might seem a bit intimidating, and you should go ahead and try this out for yourself before checking back in with me if you get stuck anywhere or if you just want to check that you got the correct answer. But I'm going to go ahead and break this problem down for you fully step by step. So let's go ahead and jump in here. Now in finding our derivative for this function, I recognize that there are a couple of different things happening here.
There is division happening between these two functions, which tells me that I need to use the quotient rule. But within my numerator and my denominator, I have a function inside of a function for both of these, telling me that at some point I'm going to also need to use the chain rule. So let's go ahead and break this down. Now in finding my derivative , we know we need to use the quotient rule, and the quotient rule tells us is our derivative. So let's go ahead and write this out using our little tool there.
So that's my low function, . is the derivative of that top function. I'm not going to take that derivative yet. I'm just going to write it out. of .
So that's my first term, minus high, my high function, . , that's the derivative of that bottom function, of . Then all of that gets divided by the square of what's below. Now, it's important here that you remember that this is already squared in our original function. But we need to square it again because it's the square of what's below.
So we need to take that full term and square it as well. So don't forget that can be easy to mess up because you already have that squared there, but we have squared it. So let's go ahead and actually take our derivatives here because we've set up our quotient rule, and now it remains to take these derivatives. Looking at this, I have 2 derivatives that I need to take, of and of . For both of these terms, I'm going to have to use the chain rule.
Now, remember that when doing these problems, it's going to be essential to stay organized, make sure that you have enough space on your page to fully work it out, and make sure that you are carrying down all of the terms. So first, I have this . I don't need to do anything there. And then I'm going to multiply that by now taking the derivative of this to the power of 4. Now, using the chain rule here, I see that I have this inside function, .
And I have this outside function because all of that is raised to the power of 4. Now, because that is raised to the power of 4, I'm going to start there with my outside function, starting from the outside, working our way in. So I'm going to go ahead and pull that 4 out to the front, decrease that power by 1 using my power rule. This becomes 4. I'm going to keep all of this in brackets just to keep it separated right now.
Four times all of that stuff on the inside, which is . Remember, when using the chain rule, we're treating that inside function like it's a variable until we get to our next step here. So , 4 times to the power of , which is 3. Now from here, we are not done using the chain rule. We are working our way inside.
Coming to that inside function, , the derivative of that inside function is just going to be 2. So all of that is just multiplied by 2, and that is that first derivative there. But we are not done. We still have a lot more terms to go. So I'm subtracting here , making sure to carry everything down that I need to.
And now here, we're going to take another derivative using the chain rule. Now here, my inside function is . My outside function is all of that stuff squared. So I'm going to go ahead and pull that 2 out to the front using the power rule, decrease that power by 1. That's going to give me 2 times, that stuff in the middle, .
Now that would just be the power of 1, so I don't need to write that because it already is raised to the power of 1. Now from here, we are, of course, not done. We know that by now. . We're going to take the derivative of that inside function.
We are working our way inside. The derivative of because I have a constant and I have , it's just going to be 1. So that is our full derivative here. Now remember not to drop your denominator. We still have a denominator here.
And we can go ahead and do a little bit of simplification because we know that if we take something squared, we raise it again to the power of 2. It is now going to be to the power of 4. So this is . Now we can do some simplification. We've taken all the derivatives that we need to.
We know that sometimes the most annoying part of the quotient rule is doing all of that algebraic simplification. But what can we do here? So looking at everything that's going on, I know that I can go ahead and take care of these constants and multiply them out, pull them out to the front. And the other thing that I do see happening here is that I have some similar terms. So I have , and I have right here.
And on the bottom, I have raised to the power of 4, which tells me that I can do a little bit of canceling here to make this a little bit cleaner. So what all can we cancel? Well, since here I have this to the lowest power, that's all that we can cancel. We can cancel one . So that means that if I cancel 1 of these out of this term, this 2 is just going to become a 1, which is not something that we're going to have to worry about writing.
And then this 4, since I can only cancel one of them, is going to become a 3. So let's go ahead and do some rewriting here. Since I have this to the power of 1, that's just , that is multiplying this 2 times 4. Remember, we are going to go ahead and multiply those constants together. That is 8 times cubed.
Double-check that we have accounted for everything going on in this term. Now I'm going to subtract here times 2. Because, remember, this term fully canceled out. So this looks a little bit nicer here. Then all of that is being divided by .
Now from here, the only thing that I want to do here is pull those constants out to the front. You could do a little bit more factoring here because are common terms happening here. But factoring that out won't simplify this much because we're still going to have the same number of terms. If factoring is something that your instructor wants you to do every single time, no matter if it really makes it a simpler term or not, that's fine. You can go ahead and do that as well.
But all I'm going to do here is pull out my constants to the front to organize this a little bit better. So this becomes 8 times times cubed minus, pulling that 2 out to the front, 2 times to the power of 4. So it looks a little bit nicer here. And then I have that on the bottom, and this is my final answer for my derivative. It is a rather long and tedious process when you have to use the quotient rule already.
But now combining here the quotient rule with the chain rule, it can get a little bit messy. Remember to take your time on these problems and double-check that you are accounting for every single term, every single derivative. Thanks for watching. Let me know if you have questions.