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Let 𝑔 of 𝑥 equal negative three 𝑓 of 𝑥 multiplied by ℎ of 𝑥 minus one.
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If the first derivative of 𝑓 of 𝑥 when 𝑥 is negative four is equal to negative one, if the first derivative of ℎ of 𝑥 when 𝑥 is equal to negative four is equal to negative nine, and ℎ of negative four is equal to negative six and 𝑓 of negative four is equal to negative one, find the derivative of 𝑔 of 𝑥 when 𝑥 is equal to negative four.
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So when we first take a look at this question, it looks quite complicated because we’ve got lots of different functions.
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But that’s the key.
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We have lots of different functions.
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So if we look at 𝑔 of 𝑥, it’s equal to negative three 𝑓 of 𝑥 multiplied by ℎ of 𝑥 minus one.
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All what we’ve got is a function multiplied by another function.
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So, therefore, we can use the product rule.
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So what the product rule tells us is that if we have 𝑦 is equal to 𝑢𝑣, so we have two things multiplied together, then the derivative of 𝑦 is gonna be equal to 𝑢 d𝑣 d𝑥 plus 𝑣 d𝑢 d𝑥.
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So that’s 𝑢 multiplied by the derivative of 𝑣 and 𝑣 multiplied by the derivative of 𝑢.
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So if we take a look at what we’ve got, we’re gonna call negative three 𝑓 of 𝑥 𝑢 and ℎ of 𝑥 minus one 𝑣.
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So, therefore, if we put these into our product rule, we’re gonna have negative three 𝑓 of 𝑥, and that’s because that’s our 𝑢, multiplied by the derivative of ℎ of 𝑥 minus one.
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Well, the derivative of ℎ of 𝑥 minus one is just gonna be the derivative of ℎ of 𝑥.
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And that’s because if you differentiate negative one, it will just become zero.
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So then we’re gonna add to this ℎ of 𝑥 minus one because this is our 𝑣.
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And then this is gonna be multiplied by negative three multiplied by the derivative of 𝑓 of 𝑥.
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We get that because it’s the derivative of 𝑢.
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And if we’ve got a constant negative three, this isn’t effective by our derivative.
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So we just do negative three multiplied by the derivative of 𝑓 of 𝑥.
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Well, we’ve done this.
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But how does this help?
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So how does this help?
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Well, this helps because we’re trying to find the derivative of 𝑔 of 𝑥 when 𝑥 is equal to negative four.
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And, therefore, the question has given us many values that we can actually substitute in.
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So I’ve now rewritten it with negative four instead of 𝑥.
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And as I said, we’ve got values in the question that we can now substitute in.
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So first of all, we’ve got 𝑓 of negative four is equal to negative one.
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So if we substitute that in we’re gonna have negative three multiplied by negative one.
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And then this is gonna be multiplied by negative nine.
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And that’s because the derivative of ℎ of 𝑥 when 𝑥 is equal to negative four is equal to negative nine.
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And this is gonna be plus negative six minus one.
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And that’s because we know that ℎ of negative four is equal to negative six which again is gonna be multiplied by the negative three multiplied by negative one.
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And this is the same as the first term that we found.
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Okay, so we’ve now got these.
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We can actually calculate it to find out what the value is going to be.
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So we’re gonna get three, and that’s cause negative three multiplied by negative one and negative multiplied by a negative is a positive, multiplied by negative nine plus negative seven.
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That’s because we had negative six minus one.
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So that’s negative seven multiplied by three.
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And that’s again because we have negative three multiplied by negative one which gives us negative 27 plus negative 21.
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Well, if we’re gonna add a negative, the same as subtracting, so this gives us negative 27 minus 21.
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So, therefore, we can say that the derivative of 𝑔 of 𝑥 when 𝑥 is equal to negative four is gonna be equal to negative 48.