Differentiating rational functions | Derivative rules | AP Calculus AB | Khan Academy

Let's say that Y is equal to five minus three X over x squared plus three X, and we want to figure out what is the derivative of Y with respect to X. Now, it might immediately jump out at you that look-look, Y is being defined as a rational expression here, as the quotient of two different expressions.

We could even view this as two different functions. You could view this one up here as U of X, so you could say this is the same thing. This is the same thing as U of X over— you could view the one in the denominator as V of X. So that one right there is V of X.

If you're taking the derivative of something that can be expressed in this way, as the quotient of two different functions, well then you could use the quotient rule. And I'll give you my little aside like I always do: the quotient rule, if you ever forget it, can be derived from the product rule, and we have videos there because the product rule is a little bit easier to remember.

But what I can do is just say, look, dy/dx, if Y is just U of X over V of X, I'm just going to restate the quotient rule. This is going to be the derivative of the function in the numerator, so d/dx of U of X times the function in the denominator times V of X, minus the function in the numerator U of X times the derivative of the function in the denominator times d/dx V of X, and we're almost there.

And then over the function in the denominator squared, the function in the denominator squared. So this might look messy, but all we have to do now is think about, well, what is the derivative of U of X? What is the derivative of V of X? And we should just be able to substitute those things back into this expression we just wrote down.

So let's do that. The derivative with respect to X of U of X is equal to, let's see, 5 minus 3x. So the derivative of 5 is 0, the derivative of negative 3x, well, that's just going to be negative 3. That's just negative 3. If any of that looks completely unfamiliar to you, I encourage you to review the derivative properties and maybe the power rule.

And now let's think about what is the derivative with respect to X, derivative with respect to X of V of X? Well, the derivative of x squared, we just bring that exponent out front. It's going to be 2 times X to the 2 minus 1, or 2 X to the first power, or just 2 X. And then the derivative of 3x is just 3. So, 2x plus 3.

Now we know everything we need to substitute back in here. The derivative of U with respect to X: this right over here is just negative 3. V of X, this we know, is x squared plus 3x. We know that this right over here is V of X, and then U of X we know is 5 minus 3x.

The derivative of V with respect to X we know is 2x plus 3, and then finally V of X we know is x squared plus 3x. So this is x squared plus 3x. And so what do we get? Well, we are going to get— and it's going to look a little bit hairy.

It's going to be equal to negative—I’ll focus. So first, we have this business up here, so it's negative 3 times x squared plus 3x. So I'm just going to distribute the negative 3. So it's negative 3x squared minus 9x.

And then from that, we are going to subtract the product of these two expressions. So let's see, what is that going to be? Well, we have a 5 times 2x which is 10x, of 5 times 3 which is 15. We have a negative 3x times 2x, so that is going to be negative 6x squared, and then a negative 3x times 3, so negative 9x.

And let's see, we can simplify that a little bit. 10x minus 9x, well that's just going to leave us with X. If we take 10x minus 9x, it's just going to be X.

And then in our denominator, we're almost there. In our denominator, we could just write that as x squared plus 3x squared, or if we want, we could expand it out. I'll just leave it like that: x squared plus 3x squared.

And so if we want to, let's just simplify or attempt to simplify this a little bit. It's going to be negative 3x squared minus 9x, and then let's see, you're going to have a negative minus X, minus X, and then minus 15, and then minus negative 6x squared, so plus 6x squared—all of that over x squared plus 3x squared or x squared plus 3x squared, I should say it that way.

Now let's see, this numerator I can simplify a little bit. Negative 3x squared plus 6x squared, that's going to be positive 3x squared.

And then, we have, other than orange, we have negative 9x minus an X, well that's going to be minus 10x, minus 10x. And then we have minus 15, so minus 15.

And so there you have it. We finally have finished. This is all going to be equal to—this is all going to be equal to 3x squared minus 10x minus 15 over x squared plus 3x squared, and we are done.