WEBVTT
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Find the first derivative of the function π¦ is equal to π to the power of π₯ times the sin of seven π₯.
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Weβre told that π¦ is equal to π to the power of π₯ times the sin of seven π₯.
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We need to find the first derivative of this expression for π¦.
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And since π¦ is a function of π₯, this means we need to find the derivative of π¦ with respect to π₯.
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We can do this by noticing that π¦ is the product of two functions.
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Itβs π to the power of π₯ multiplied by the sin of seven π₯.
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And in fact, itβs the product of two differentiable functions.
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We know how to find their derivatives.
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So we can differentiate this by using the product rule.
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We recall the product rule tells us for differentiable functions π of π₯ and π of π₯, the derivative of π of π₯ multiplied by π of π₯ with respect to π₯ is equal to π prime of π₯ times π of π₯ plus π prime of π₯ times π of π₯.
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In our case, we want to find the first derivative of π¦ with respect to π₯.
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Thatβs the derivative of π to the power of π₯ multiplied by the sin of seven π₯ with respect to π₯.
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So we can do this by using the product rule.
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We need to set π of π₯ equal to π to the power of π₯ and π of π₯ to be equal to the sin of seven π₯.
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This means weβre going to need to find expressions for π prime of π₯ and π prime of π₯.
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Letβs start with π prime of π₯.
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Thatβs the derivative of π to the power of π₯ with respect to π₯.
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And we can do this by recalling the derivative of the exponential function π to the power of π₯ is just equal to itself, π to the power of π₯.
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So π prime of π₯ is just equal to π to the power of π₯.
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We now need to find an expression for π prime of π₯.
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Thatβs the derivative of the sin of seven π₯ with respect to π₯.
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And to do this, we need to recall one of our standard trigonometric derivative results.
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For any real constant π, the derivative of the sin of ππ₯ with respect to π₯ is equal to π times the cos of ππ₯.
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In this case, the value of π is equal to seven.
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So we get π prime of π₯ is equal to seven times the cos of seven π₯.
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Weβre now ready to find the first derivative of π¦ with respect to π₯ by using the product rule.
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Itβs equal to π prime of π₯ times π of π₯ plus π prime of π₯ times π of π₯.
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Substituting in our expressions for π of π₯, π of π₯, π prime of π₯, and π prime of π₯, we get that π¦ prime is equal to π to the power of π₯ times the sin of seven π₯ plus seven times the cos of seven π₯ multiplied by π to the power of π₯.
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And we could leave our answer like this or we could take out the common factor of π to the power of π₯.
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However, instead, weβll leave this as two terms, and weβll write π to the power of π₯ at the front of our second term.
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We do this to avoid confusion of the cos of seven π₯ all multiplied by π to the power of π₯ with the cos of seven π₯ times π to the power of π₯.
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And this gives us our final answer.
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Therefore, given π¦ is equal to π to the power of π₯ times the sin of seven π₯, we were able to find the first derivative of π¦ with respect to π₯ by using the product rule.
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We got that π¦ prime was equal to π to the power of π₯ times the sin of seven π₯ plus seven π to the power of π₯ multiplied by the cos of seven π₯.