### Video Transcript

Find the first derivative of the

function 𝑦 equals 𝑥 to the power of four multiplied by four 𝑥 plus nine to the

power of nine at 𝑥 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 𝑦 equals 𝑢𝑣. So therefore, we’re actually gonna

use the product rule to help us differentiate it. And the product rule states that

𝑑𝑦 𝑑𝑥 is equal to 𝑢 𝑑𝑣 𝑑𝑥 plus 𝑣 𝑑𝑢 𝑑𝑥.

So first of all, we pick out what’s

gonna be our 𝑢 and our 𝑣. So our 𝑢 is going to be 𝑥 to the

power of four and our 𝑣 is equal to four 𝑥 plus nine to the power of nine. So now, what we want to do is

actually find out what 𝑑𝑢 𝑑𝑥 and 𝑑𝑣 𝑑𝑥 actually are. So I’m gonna start with 𝑑𝑢 𝑑𝑥

and 𝑑𝑢 𝑑𝑥 is gonna be equal to four 𝑥 cubed. And that’s because we’ve obviously

differentiated our 𝑥 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 — so four multiplied by one. And then what we did is we actually

subtracted one from our exponent — so four minus one. So that gave us four 𝑥 to the

power of three.

Okay, great, so now let’s move on

and differentiate 𝑣 to find 𝑑𝑣 𝑑𝑥. Well, now to find 𝑑𝑣 𝑑𝑥, what

we need to do is we actually need to use a rule to help us actually differentiate

four 𝑥 plus nine to the power of nine. And the rule we’re actually gonna

use is the chain rule. And the chain rule actually tells

us that 𝑑𝑦 𝑑𝑥 is equal to 𝑑𝑦 𝑑𝑢 multiplied by 𝑑𝑢 𝑑𝑥.

Okay, so now we have this rule,

let’s look at our term and see if we can apply it. Well, first of all, we’re gonna

have a look at 𝑢 and say that it’s equal to four 𝑥 plus nine. So therefore, 𝑑𝑢 𝑑𝑥 is just

gonna be four because if you differentiate four 𝑥, you get four and if you

differentiate nine, you just get zero.

Okay, so now we can move on to

𝑦. Well, 𝑦 would be equal to 𝑢 to

the power of nine. So therefore, 𝑑𝑦 𝑑𝑢 is equal to

nine 𝑢 to the power of eight. So then if we apply the chain rule,

we can say that 𝑑𝑦 𝑑𝑥 is equal to four multiplied by nine 𝑢 to the power of

eight, which is equal to 36𝑢 to the power of eight. So then, we just substitute back in

that 𝑢 is equal to four 𝑥 plus nine and we get that 𝑑𝑦 𝑑𝑥 is equal to 36

multiplied by four 𝑥 plus nine to the power of eight. So therefore, we can say the 𝑑𝑣

𝑑𝑥 is equal to 36 multiplied by four 𝑥 plus nine to the power of eight.

So therefore, now that we found our

𝑑𝑣 𝑑𝑥 and our 𝑑𝑢 𝑑𝑥, what we can actually do is apply our product rule to

find 𝑑𝑦 𝑑𝑥. So the first derivative of our

function 𝑥 to the power of four multiplied by four 𝑥 plus nine to the power of

nine. So therefore, we get 𝑑𝑦 𝑑𝑥 is

equal to 𝑥 to the power of four because that’s our 𝑢 and then multiplied by our

𝑑𝑣 𝑑𝑥 which is 36 multiplied by four 𝑥 plus nine to the power of eight plus our

𝑣 which is four 𝑥 plus nine to the power of nine multiplied by our 𝑑𝑢 𝑑𝑥 which

is four 𝑥 cubed.

So great, we’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 𝑥 is equal to

negative two. And in order to do this, we need to

substitute in 𝑥 is equal to negative two into our first derivative.

So we’re gonna get that the first

derivative with negative two substituted in for 𝑥 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 𝑦 equals 𝑥 to the power of four multiplied by

four 𝑥 plus nine to the power of nine at 𝑥 equals negative two is equal to

544.