WEBVTT
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If 𝑦 equals two sin three plus eight 𝑥, determine d𝑦 d𝑥.
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So in order to determine d𝑦 d𝑥, we’re gonna have to differentiate our function.
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And in order to do this, what we actually have to use is a couple of general rules to help us out.
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Well first of all, we’ve got that if 𝑦 is equal to sin 𝑥, then we know that d𝑦 d𝑥 is gonna be equal to cos 𝑥.
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So if we differentiate sin 𝑥 we, get cos 𝑥.
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Okay?
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This is really useful cause it helps us understand what’s gonna to happen to our sin 𝑥.
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However, if we look at our function, it’s in a slightly different form.
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Our function is more in a form like this, so we’ve got that 𝑦 is equal to 𝑎 sin 𝑓 𝑥.
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Well because if we look at our function, what we’ve got is 𝑎, so we’ve got a constant, so it’s two, and we got sin, and then we’ve got a function, so we’ve got three plus eight 𝑥.
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So if we have it in this form, we can say that d𝑦 d𝑥 gonna be equal to 𝑎 multiplied by the derivative of the function inside our function multiplied by the cosine of that function.
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Okay, great.
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So now that we’ve got this, we can go on and differentiate 𝑦 equals two sin three plus eight 𝑥.
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So we’ve got 𝑦 equals two sin three plus eight 𝑥.
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So therefore, if we now apply the rule that we looked at which was how we worked out what d𝑦 d𝑥 would be for function in this form.
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And actually, by the way, this actually comes from the chain rule, so we’ve actually used the chain rule to get this.
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We’re gonna get d𝑦 d𝑥 is equal to two multiplied by derivative of three plus eight 𝑥 with respect to 𝑥 multiplied by the cosine of three plus eight 𝑥, which is gonna be equal to two multiplied by eight multiplied by the cosine of three plus eight 𝑥.
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And we actually got this because if we differentiate with respect to 𝑥 three plus eight 𝑥, what we’re actually gonna get is zero plus eight, which is just eight.
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That’s because if we differentiate an integer, we just get zero.
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And if we differentiate eight 𝑥, we got eight multiplied by the exponent.
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So that’s gonna be eight multiplied by one, so that just gives us eight.
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And then it’s 𝑥 to the power of — then you subtract one from the exponent, so one minus one, which would be 𝑥 to the power of zero, which would just give us one, so we’re just left with eight.
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Okay, great!
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so one more step, and then we can find our d𝑦 d𝑥.
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So therefore, if we multiply two by eight, we get 16.
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So we can say that if 𝑦 equals two sin three plus eight 𝑥, d𝑦 d𝑥 is gonna be equal to 16 cos three plus eight 𝑥.