Question Video: Finding and Evaluating the First Derivative of Polynomial Functions Using the Chain Rule and the Product Rule | Nagwa Question Video: Finding and Evaluating the First Derivative of Polynomial Functions Using the Chain Rule and the Product Rule | Nagwa

# Question Video: Finding and Evaluating the First Derivative of Polynomial Functions Using the Chain Rule and the Product Rule Mathematics • Second Year of Secondary School

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Find the first derivative of the function π¦ = π₯β΄ (4π₯ + 9)βΉ at π₯ = β2.

04:24

### Video Transcript

Find the first derivative of the function π¦ equals π₯ to the power of four multiplied by four π₯ plus nine to the power of nine at π₯ equals negative two.

Well, the first thing we need to do to actually solve this problem is actually differentiate our function. Well, we can see that our function is in the form π¦ equals π’π£. So therefore, weβre actually gonna use the product rule to help us differentiate it. And the product rule states that ππ¦ ππ₯ is equal to π’ ππ£ ππ₯ plus π£ ππ’ ππ₯.

So first of all, we pick out whatβs gonna be our π’ and our π£. So our π’ is going to be π₯ to the power of four and our π£ is equal to four π₯ plus nine to the power of nine. So now, what we want to do is actually find out what ππ’ ππ₯ and ππ£ ππ₯ actually are. So Iβm gonna start with ππ’ ππ₯ and ππ’ ππ₯ is gonna be equal to four π₯ cubed. And thatβs because weβve obviously differentiated our π₯ to the power of four.

And just to remind us how we actually did that, what we did is we actually multiplied the coefficient of our term by the exponent β so four multiplied by one. And then what we did is we actually subtracted one from our exponent β so four minus one. So that gave us four π₯ to the power of three.

Okay, great, so now letβs move on and differentiate π£ to find ππ£ ππ₯. Well, now to find ππ£ ππ₯, what we need to do is we actually need to use a rule to help us actually differentiate four π₯ plus nine to the power of nine. And the rule weβre actually gonna use is the chain rule. And the chain rule actually tells us that ππ¦ ππ₯ is equal to ππ¦ ππ’ multiplied by ππ’ ππ₯.

Okay, so now we have this rule, letβs look at our term and see if we can apply it. Well, first of all, weβre gonna have a look at π’ and say that itβs equal to four π₯ plus nine. So therefore, ππ’ ππ₯ is just gonna be four because if you differentiate four π₯, you get four and if you differentiate nine, you just get zero.

Okay, so now we can move on to π¦. Well, π¦ would be equal to π’ to the power of nine. So therefore, ππ¦ ππ’ is equal to nine π’ to the power of eight. So then if we apply the chain rule, we can say that ππ¦ ππ₯ is equal to four multiplied by nine π’ to the power of eight, which is equal to 36π’ to the power of eight. So then, we just substitute back in that π’ is equal to four π₯ plus nine and we get that ππ¦ ππ₯ is equal to 36 multiplied by four π₯ plus nine to the power of eight. So therefore, we can say the ππ£ ππ₯ is equal to 36 multiplied by four π₯ plus nine to the power of eight.

So therefore, now that we found our ππ£ ππ₯ and our ππ’ ππ₯, what we can actually do is apply our product rule to find ππ¦ ππ₯. So the first derivative of our function π₯ to the power of four multiplied by four π₯ plus nine to the power of nine. So therefore, we get ππ¦ ππ₯ is equal to π₯ to the power of four because thatβs our π’ and then multiplied by our ππ£ ππ₯ which is 36 multiplied by four π₯ plus nine to the power of eight plus our π£ which is four π₯ plus nine to the power of nine multiplied by our ππ’ ππ₯ which is four π₯ cubed.

So great, weβre actually at the stage where we found the first derivative of our function. But what do we do now? So now, what we need to do is actually look at what the first derivative is going to be when π₯ is equal to negative two. And in order to do this, we need to substitute in π₯ is equal to negative two into our first derivative.

So weβre gonna get that the first derivative with negative two substituted in for π₯ is equal to negative two to the power of four multiplied by 36 times four times negative two plus nine to the power of eight plus four multiplied by negative two plus nine to the power of nine multiplied by four multiplied by negative two cubed which is gonna be equal to 16 multiplied by 36 plus four multiplied by negative eight which is equal to 544.

So therefore, we can say that the first derivative of the function π¦ equals π₯ to the power of four multiplied by four π₯ plus nine to the power of nine at π₯ equals negative two is equal to 544.

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