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Use the fundamental theorem of calculus to find the derivative of the function 𝑔 of 𝑥, which is equal to the integral between three and 𝑥 of the natural log of one plus 𝑡 to the power of five with respect to 𝑡.
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For this question, know that we’ve been given a function 𝑔 of 𝑥, which is defined by an integral.
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We then been asked to find the derivative of this function.
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Now, our first thought might be to try and differentiate the integral with standard techniques and then to differentiate with respect to 𝑥.
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Here, this would be a mistake, since the integral we’ve been given would probably be messy and difficult to tackle.
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Instead, the question gives us a hint that we should be using the fundamental theorem of calculus, which we’ll be abbreviating to FTC.
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Specifically, the first part of the theorem tells us that if 𝑓 is a continuous function on the closed interval between 𝑎 and 𝑏 and capital 𝐹 of 𝑥 is defined by the integral between 𝑎 and 𝑥 of 𝑓 of 𝑡 with respect to 𝑡.
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Then 𝐹 prime of 𝑥 is equal to 𝑓 of 𝑥 for all values of 𝑥 on the open interval between 𝑎 and 𝑏.
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This is an incredibly powerful theorem and we can understand its meaning by applying it to our question.
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Indeed, we know that the function we’ve been given in the question does match the form of the fundamental theorem of calculus with 𝑔 of 𝑥 representing capital 𝐹 of 𝑥, the natural log of one plus 𝑡 to the power of five representing lowercase 𝑓 of 𝑡, the lower limit of our integration three being the constant 𝑎, and of course, the upper limit being 𝑥.
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Given the forms match, we can directly use the fundamental theorem of calculus to reach a result for 𝑔 prime of 𝑥, which here represents capital 𝐹 prime of 𝑥.
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We know the function lowercase 𝑓 of 𝑡 and so to find lowercase 𝑓 of 𝑥, we simply replace the 𝑡s by 𝑥s.
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This means lowercase 𝑓 of 𝑥 is equal to the natural log of one plus 𝑥 to the power of five.
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And in fact, we’ve already reached our answer for 𝑔 prime of 𝑥.