# Question Video: Understanding the Product Rule Mathematics • Higher Education

Let ๐(๐ฅ) = โ3๐(๐ฅ) [โ(๐ฅ) โ 1]. If ๐โฒ(โ4) = โ1, โโฒ(โ4) = โ9, โ(โ4) = โ6, and ๐(โ4) = โ1, find ๐โฒ(โ4).

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### Video Transcript

Let ๐ of ๐ฅ equal negative three ๐ of ๐ฅ multiplied by โ of ๐ฅ minus one. If the first derivative of ๐ of ๐ฅ when ๐ฅ is negative four is equal to negative one, if the first derivative of โ of ๐ฅ when ๐ฅ is equal to negative four is equal to negative nine, and โ of negative four is equal to negative six and ๐ of negative four is equal to negative one, find the derivative of ๐ of ๐ฅ when ๐ฅ is equal to negative four.

So when we first take a look at this question, it looks quite complicated because weโve got lots of different functions. But thatโs the key. We have lots of different functions. So if we look at ๐ of ๐ฅ, itโs equal to negative three ๐ of ๐ฅ multiplied by โ of ๐ฅ minus one. All what weโve got is a function multiplied by another function. So, therefore, we can use the product rule.

So what the product rule tells us is that if we have ๐ฆ is equal to ๐ข๐ฃ, so we have two things multiplied together, then the derivative of ๐ฆ is gonna be equal to ๐ข d๐ฃ d๐ฅ plus ๐ฃ d๐ข d๐ฅ. So thatโs ๐ข multiplied by the derivative of ๐ฃ and ๐ฃ multiplied by the derivative of ๐ข. So if we take a look at what weโve got, weโre gonna call negative three ๐ of ๐ฅ ๐ข and โ of ๐ฅ minus one ๐ฃ.

So, therefore, if we put these into our product rule, weโre gonna have negative three ๐ of ๐ฅ, and thatโs because thatโs our ๐ข, multiplied by the derivative of โ of ๐ฅ minus one. Well, the derivative of โ of ๐ฅ minus one is just gonna be the derivative of โ of ๐ฅ. And thatโs because if you differentiate negative one, it will just become zero. So then weโre gonna add to this โ of ๐ฅ minus one because this is our ๐ฃ. And then this is gonna be multiplied by negative three multiplied by the derivative of ๐ of ๐ฅ. We get that because itโs the derivative of ๐ข. And if weโve got a constant negative three, this isnโt effective by our derivative. So we just do negative three multiplied by the derivative of ๐ of ๐ฅ.

Well, weโve done this. But how does this help? So how does this help? Well, this helps because weโre trying to find the derivative of ๐ of ๐ฅ when ๐ฅ is equal to negative four. And, therefore, the question has given us many values that we can actually substitute in. So Iโve now rewritten it with negative four instead of ๐ฅ. And as I said, weโve got values in the question that we can now substitute in.

So first of all, weโve got ๐ of negative four is equal to negative one. So if we substitute that in weโre gonna have negative three multiplied by negative one. And then this is gonna be multiplied by negative nine. And thatโs because the derivative of โ of ๐ฅ when ๐ฅ is equal to negative four is equal to negative nine. And this is gonna be plus negative six minus one. And thatโs because we know that โ of negative four is equal to negative six which again is gonna be multiplied by the negative three multiplied by negative one. And this is the same as the first term that we found.

Okay, so weโve now got these. We can actually calculate it to find out what the value is going to be. So weโre gonna get three, and thatโs cause negative three multiplied by negative one and negative multiplied by a negative is a positive, multiplied by negative nine plus negative seven. Thatโs because we had negative six minus one. So thatโs negative seven multiplied by three. And thatโs again because we have negative three multiplied by negative one which gives us negative 27 plus negative 21. Well, if weโre gonna add a negative, the same as subtracting, so this gives us negative 27 minus 21. So, therefore, we can say that the derivative of ๐ of ๐ฅ when ๐ฅ is equal to negative four is gonna be equal to negative 48.
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