# Video: Differentiating Rational Functions Using the Quotient Rule

Find ππ¦/ππ₯, given that π¦ = (π₯Β³ + 7π₯Β² + 6)/(π₯ + 8).

04:47

### Video Transcript

Find ππ¦ ππ₯, given that π¦ equals π₯ cubed plus seven π₯ squared plus six over π₯ plus eight.

Now, in order to actually solve this problem, what weβre gonna have to do is differentiate our function. But to enable us to do that, what weβre gonna use is the quotient rule. And we can use the quotient rule because our function is in the form π’ over π£. So as weβve got that, what weβre gonna have a look at is what the quotient rule actually is. Well, the quotient rule tells us that ππ¦ ππ₯ is equal to π£ ππ’ ππ₯ minus π’ ππ£ ππ₯ over π£ squared. So what this actually means in practice is π£ multiplied by the derivative of π’ minus π’ multiplied by the derivative of π£ all over π£ squared.

Okay, great. So now, weβve got the quotient rule. Letβs apply it to actually find ππ¦ ππ₯. So the first thing Iβve done is Iβve actually identified π’ and π£ in our question. So our π’ is gonna be π₯ cubed plus seven π₯ squared plus six because thatβs our numerator. And our π£ is gonna be our denominator which is π₯ plus eight. So next, what Iβm gonna do, Iβm actually gonna find ππ’ ππ₯. So Iβm gonna differentiate our π’ value weβve got here. So weβll differentiate π₯ cubed plus seven π₯ squared plus six.

Just a quickly recap, how weβre gonna differentiate each of these. What weβre gonna look at is the differentiations of general rules, so our power rule. And what it is, is if weβve got a function thatβs in the form ππ₯ to the power of π, then the first derivative of that function is gonna be equal to πππ₯ to the power of π minus one. So our coefficient multiplied by our exponent. And then, we reduce the exponent by one because itβs gonna be πππ₯ to the power of π minus one. Okay, weβve just recapped that. Letβs move on and differentiate and find ππ’ ππ₯.

So we go ahead and apply that power rule. And our first term is gonna be three π₯ squared. And thatβs because our coefficient is one multiplied by the exponent three gives us three. And then we reduce the exponent by one, so π₯ to the power of two or π₯ squared. And our second term is just 14π₯, so positive 14π₯. The six, when we differentiate the six, it actually disappears cause if you differentiate just an integer on its own, we get zero. Okay, great. So weβve now found ππ’ ππ₯.

So now, we move on to ππ£ ππ₯. So again, we differentiate π₯ plus eight. And when we do that, we just get one. And thatβs because π₯ differentiates to one. And as before, the positive eight just differentiates to zero. Great, so weβve now found ππ’ ππ₯ and ππ£ ππ₯. So now, we can move on to the final stage which is to actually apply the quotient rule to find out ππ¦ ππ₯.

So when we actually apply the quotient rule, weβre gonna get ππ¦ ππ₯ is equal to. And then first of all, π£ ππ’ ππ₯. So our π£, π₯ plus eight, multiplied by our ππ’ ππ₯, which is three π₯ squared plus 14π₯. And then weβre gonna subtract π₯ cubed plus seven π₯ squared plus six because thatβs our π’ ππ£ ππ₯. And itβs just that value there because our ππ£ ππ₯ is just one. Okay, great. And then finally, what we do is we actually divide by π£ squared. So itβs π₯ plus eight squared. Okay, fabulous. Weβre actually at the point now where weβve put all of our values into our quotient rule. Letβs simplify to find ππ¦ ππ₯.

Well, our first stage is to actually expand the parentheses. So weβre gonna have π₯ multiplied by three π₯ squared which is gonna give us three π₯ cubed. And then, we have π₯ multiplied by positive 14π₯ which gives us plus 14π₯ squared. And then next, we have positive eight multiplied by three π₯ squared which gives us positive 24π₯ squared. So finally, we get plus 112π₯. And thatβs because we had positive eight multiplied by positive 14π₯. Then we have minus π₯ cubed minus seven π₯ squared minus six.

So at this point, we actually can draw your attention to a possible common mistake. So as you can see, thereβs a negative in front of the parentheses. I put the second part in a parentheses because actually just to remind you that actually everything inside there has to be negative. So itβs negative π₯ cubed minus seven π₯ squared minus six. Because commonly, people will just do minus π₯ cubed. But then, theyβll just do plus seven π₯ squared plus six. So be careful for that. Okay, and then this is all divided by π₯ plus eight squared.

And when we simplify this, we have three π₯ cubed minus π₯ cubed which gives us two π₯ cubed. Then we have 14π₯ squared plus 24π₯ squared minus seven π₯ squared which gives us positive 31π₯ squared. Then we have plus 112π₯ minus six over π₯ plus eight all squared.

So therefore, we can say that given that π¦ is equal to π₯ cubed plus seven π₯ squared plus six over π₯ plus eight, then ππ¦ ππ₯ is equal to two π₯ cubed plus 31π₯ squared plus 112π₯ minus six over π₯ plus eight all squared.