Derivative of 4x || Power Rule of Differentiation
Derivative of 4x || Power Rule of Differentiation

### 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.

You are watching: Question Video: Finding and Evaluating the First Derivative of Polynomial Functions Using the Chain Rule and the Product Rule. Info created by Bút Chì Xanh selection and synthesis along with other related topics.