Question Video: Understanding the Product Rule Mathematics • Higher Education

Let ๐‘”(๐‘ฅ) = โˆ’3๐‘“(๐‘ฅ) [โ„Ž(๐‘ฅ) โˆ’ 1]. If ๐‘“โ€ฒ(โˆ’4) = โˆ’1, โ„Žโ€ฒ(โˆ’4) = โˆ’9, โ„Ž(โˆ’4) = โˆ’6, and ๐‘“(โˆ’4) = โˆ’1, find ๐‘”โ€ฒ(โˆ’4).

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Video Transcript

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

So when we first take a look at this question, it looks quite complicated because weโ€™ve got lots of different functions. But thatโ€™s the key. We have lots of different functions. So if we look at ๐‘” of ๐‘ฅ, itโ€™s equal to negative three ๐‘“ of ๐‘ฅ multiplied by โ„Ž of ๐‘ฅ minus one. All what weโ€™ve 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 ๐‘ฆ is equal to ๐‘ข๐‘ฃ, so we have two things multiplied together, then the derivative of ๐‘ฆ is gonna be equal to ๐‘ข d๐‘ฃ d๐‘ฅ plus ๐‘ฃ d๐‘ข d๐‘ฅ. So thatโ€™s ๐‘ข multiplied by the derivative of ๐‘ฃ and ๐‘ฃ multiplied by the derivative of ๐‘ข. So if we take a look at what weโ€™ve got, weโ€™re gonna call negative three ๐‘“ of ๐‘ฅ ๐‘ข and โ„Ž of ๐‘ฅ minus one ๐‘ฃ.

So, therefore, if we put these into our product rule, weโ€™re gonna have negative three ๐‘“ of ๐‘ฅ, and thatโ€™s because thatโ€™s our ๐‘ข, multiplied by the derivative of โ„Ž of ๐‘ฅ minus one. Well, the derivative of โ„Ž of ๐‘ฅ minus one is just gonna be the derivative of โ„Ž of ๐‘ฅ. And thatโ€™s because if you differentiate negative one, it will just become zero. So then weโ€™re gonna add to this โ„Ž of ๐‘ฅ minus one because this is our ๐‘ฃ. And then this is gonna be multiplied by negative three multiplied by the derivative of ๐‘“ of ๐‘ฅ. We get that because itโ€™s the derivative of ๐‘ข. And if weโ€™ve got a constant negative three, this isnโ€™t effective by our derivative. So we just do negative three multiplied by the derivative of ๐‘“ of ๐‘ฅ.

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

So first of all, weโ€™ve got ๐‘“ of negative four is equal to negative one. So if we substitute that in weโ€™re gonna have negative three multiplied by negative one. And then this is gonna be multiplied by negative nine. And thatโ€™s because the derivative of โ„Ž of ๐‘ฅ when ๐‘ฅ is equal to negative four is equal to negative nine. And this is gonna be plus negative six minus one. And thatโ€™s because we know that โ„Ž 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 weโ€™ve now got these. We can actually calculate it to find out what the value is going to be. So weโ€™re gonna get three, and thatโ€™s cause negative three multiplied by negative one and negative multiplied by a negative is a positive, multiplied by negative nine plus negative seven. Thatโ€™s because we had negative six minus one. So thatโ€™s negative seven multiplied by three. And thatโ€™s again because we have negative three multiplied by negative one which gives us negative 27 plus negative 21. Well, if weโ€™re 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 ๐‘ฅ when ๐‘ฅ is equal to negative four is gonna be equal to negative 48.

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