In this problem, we're asked to find the derivative of the function y equals (x+1)(x²-3x) over x³. Now you should go ahead if you haven't already and try to solve this problem on your own because you already have all of the skills that you need to do so. But I'm going to go ahead and jump into finding this derivative here, a dy/dx of my function. Now the first thing that I notice about my function here is that I have one function, this function on the top, being divided by another function on the bottom here, meaning I need to use the quotient rule. Now remember the quotient rule tells us "low d high minus high d low over the square of what's below."
So going ahead and applying that quotient rule here, I start with my low function, x³, and multiply it by the derivative of that top function, my high function. Now looking at this function on the top, I have (x + 1) multiplying (x² - 3x). So in order to take this derivative, I need to use the product rule within the quotient rule that I'm already using here. But we know how to do this. So let's go ahead and focus on finding this derivative of this top function here.
Now I have two functions being multiplied, that function on the left and my function on the right. Remember, our product rule tells us left d right plus right d left. That's our derivative. So I'm going to take that left-hand function first, x + 1, and multiply it by the derivative of this right-hand function using the power rule that gives me 2x - 3. Then I am adding this together with my right-hand function, x² - 3x, leaving that as is, times the derivative of that left-hand function.
The derivative of x + 1 is just one, a much easier derivative there, and that's that full derivative of that top function. So we can move on in our quotient rule here. We are subtracting from this low d high. We're subtracting high d low. So taking that high function as is, which is rather long, (x+1)(x²-3x) and multiplying that by the derivative of that low function, a little bit easier to find here using the power rule, that is 3x².
So that is my full numerator here. I'm dividing this all by the square of what's below, so that is x³ squared. And now from here, we have applied our quotient rule fully, and we just have a bunch of algebra to follow. Now because we have so much going on in our numerator because we had to use the product rule, we need to be super careful here when doing all of this algebra. So make sure that you stay organized, take your time so that you can get the correct answer here.
So let's continue on doing some algebra. Now the first thing that I'm going to do is I'm going to look at everything that I need to multiply out and focus on each of those terms. Foiling out and combining like terms carefully, we end up with a simplified polynomial. Remember, our denominator is still x to the power of 6.
Finally, we distribute negatives, combine like terms one last time, and simplify by canceling out powers of x where possible, as all terms involve x. After all the cancellations and simplifications, we arrive at dy/dx = (2x + 6) ÷ x³, which simplifies further to dy/dx = (2x + 6)/x³. I know that this is a long process, but this is our final derivative here for dy/dx of our original function.
It is (2x + 6)/x³. If you made it this far in the video, thanks so much for watching, and of course, feel free to let us know if you have questions.
