### 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, weβre actually gonna
use the product rule to help us differentiate it. And the product rule states that
ππ¦ ππ₯ is equal to π’ ππ£ ππ₯ plus π£ ππ’ ππ₯.

So first of all, we pick out whatβs
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 ππ’ ππ₯ and ππ£ ππ₯ actually are. So Iβm gonna start with ππ’ ππ₯
and ππ’ ππ₯ is gonna be equal to four π₯ cubed. And thatβs because weβve 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 letβs move on
and differentiate π£ to find ππ£ ππ₯. Well, now to find ππ£ ππ₯, 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 weβre actually gonna
use is the chain rule. And the chain rule actually tells
us that ππ¦ ππ₯ is equal to ππ¦ ππ’ multiplied by ππ’ ππ₯.

Okay, so now we have this rule,
letβs look at our term and see if we can apply it. Well, first of all, weβre gonna
have a look at π’ and say that itβs equal to four π₯ plus nine. So therefore, ππ’ ππ₯ 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, ππ¦ ππ’ is equal to
nine π’ to the power of eight. So then if we apply the chain rule,
we can say that ππ¦ ππ₯ 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 ππ¦ ππ₯ is equal to 36
multiplied by four π₯ plus nine to the power of eight. So therefore, we can say the ππ£
ππ₯ is equal to 36 multiplied by four π₯ plus nine to the power of eight.

So therefore, now that we found our
ππ£ ππ₯ and our ππ’ ππ₯, what we can actually do is apply our product rule to
find ππ¦ ππ₯. 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 ππ¦ ππ₯ is
equal to π₯ to the power of four because thatβs our π’ and then multiplied by our
ππ£ ππ₯ 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 ππ’ ππ₯ which
is four π₯ cubed.

So great, weβre 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 weβre 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.