Question Video: Differentiating Exponential Functions Using the Chain Rule Mathematics • Higher Education

Find ππ¦/ππ₯ if π¦ = 4^(9π₯Β² β 4π₯ + 8).

03:25

Video Transcript

Find ππ¦ ππ₯ if π¦ is equal to four to the power of nine π₯ squared minus four π₯ plus eight.

Now, in order to actually find ππ¦ ππ₯, what weβre gonna do is actually weβre gonna use the chain rule. And the chain rule states that ππ¦ ππ₯ is equal to ππ¦ ππ’ multiplied by ππ’ ππ₯. And this is when π¦ is equal to a function of π’ and π’ is equal to a function of π₯. Okay, so we now got the chain rule. Letβs apply it to our problem to actually solve and find ππ¦ ππ₯.

So first of all, we need to identify our π’. So thatβs gonna be nine π₯ squared minus four π₯ plus eight. And therefore, our π¦ is gonna be equal to four to the power of π’. Okay, thatβs the first step. Now, what we need to do is actually differentiate to find ππ¦ ππ’ and ππ’ ππ₯.

Iβm gonna start by finding ππ’ ππ₯ and I will do that by differentiating the expression nine π₯ squared minus four π₯ plus eight. And this is just gonna give us 18π₯ minus four. Weβve done that the normal way. So we just differentiated or highlighted that with the first term. So what we did is we multiplied the exponent by the coefficient. So two multiplied by nine is 18 and then we reduced the exponent by one β so from two to one. So we just get 18π₯.

So great, weβve differentiated that. Now, letβs move on and find ππ¦ ππ’. So now to find ππ¦ ππ’, what weβre gonna have to do is actually differentiate four to the power of π’. And in order to do this, weβre actually gonna use the general rule which is that if weβre gonna differentiate π to the power of π₯, then weβre gonna get an answer of π to the power of π₯ multiplied by the natural logarithm of π. So therefore, as our result, weβre gonna get four to power of π’ because thatβs our π and then multiply it by the natural logarithm of four because again thatβs our π because itβs π to the power of π₯ ln π.

Great, so weβve now differentiated both parts. We can use the chain rule to put them together to find ππ¦ ππ₯. So now, weβre actually gonna apply the chain rule, which tells us that ππ¦ ππ₯ is equal to ππ¦ ππ’ multiplied by ππ’ ππ₯. So weβre gonna get ππ¦ ππ₯ is equal to four to the power of π’ ln four and thatβs because that was our ππ¦ ππ’ and then multiplied by 18π₯ minus four because that was our ππ’ ππ₯.

Okay, great, but this isnβt a final answer because we can see there are actually still including π’. So what we now need to do is actually substitute in our value for π’. So therefore, weβre gonna get β and Iβve just rearranged a bit here β 18π₯ minus four multiplied by four to the power of nine π₯ squared minus four π₯ plus eight because that was our π’ and then thatβs multiplied by the natural logarithm of four.

So therefore, after some tidying up using factoring, we can say that if π¦ is equal to four to the power of nine π₯ squared minus four π₯ plus eight, therefore ππ¦ ππ₯ is gonna be equal to two multiplied by nine π₯ minus two. And we got that because actually we factored 18π₯ minus four because two is a factor of 18π₯ and negative four. And this is multiplied by four to the power of nine π₯ squared minus four π₯ plus eight ln four.