Video: Finding the Derivative of a Function Defined by an Integral

Use the fundamental theorem of calculus to find the derivative of the function ๐‘”(๐‘ฅ) = โˆซ_(3) ^(๐‘ฅ) ln (1 + ๐‘กโต) d๐‘ก.

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Video Transcript

Use the fundamental theorem of calculus to find the derivative of the function ๐‘” of ๐‘ฅ, which is equal to the integral between three and ๐‘ฅ of the natural log of one plus ๐‘ก to the power of five with respect to ๐‘ก.

For this question, know that weโ€™ve been given a function ๐‘” of ๐‘ฅ, which is defined by an integral. We then been asked to find the derivative of this function. Now, our first thought might be to try and differentiate the integral with standard techniques and then to differentiate with respect to ๐‘ฅ. Here, this would be a mistake, since the integral weโ€™ve been given would probably be messy and difficult to tackle. Instead, the question gives us a hint that we should be using the fundamental theorem of calculus, which weโ€™ll be abbreviating to FTC. Specifically, the first part of the theorem tells us that if ๐‘“ is a continuous function on the closed interval between ๐‘Ž and ๐‘ and capital ๐น of ๐‘ฅ is defined by the integral between ๐‘Ž and ๐‘ฅ of ๐‘“ of ๐‘ก with respect to ๐‘ก. Then ๐น prime of ๐‘ฅ is equal to ๐‘“ of ๐‘ฅ for all values of ๐‘ฅ on the open interval between ๐‘Ž and ๐‘.

This is an incredibly powerful theorem and we can understand its meaning by applying it to our question. Indeed, we know that the function weโ€™ve been given in the question does match the form of the fundamental theorem of calculus with ๐‘” of ๐‘ฅ representing capital ๐น of ๐‘ฅ, the natural log of one plus ๐‘ก to the power of five representing lowercase ๐‘“ of ๐‘ก, the lower limit of our integration three being the constant ๐‘Ž, and of course, the upper limit being ๐‘ฅ. Given the forms match, we can directly use the fundamental theorem of calculus to reach a result for ๐‘” prime of ๐‘ฅ, which here represents capital ๐น prime of ๐‘ฅ. We know the function lowercase ๐‘“ of ๐‘ก and so to find lowercase ๐‘“ of ๐‘ฅ, we simply replace the ๐‘กs by ๐‘ฅs. This means lowercase ๐‘“ of ๐‘ฅ is equal to the natural log of one plus ๐‘ฅ to the power of five. And in fact, weโ€™ve already reached our answer for ๐‘” prime of ๐‘ฅ.

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