### Video Transcript

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

For this question, know that weโve
been given a function ๐ of ๐ฅ, 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 ๐ฅ. Here, this would be a mistake,
since the integral weโve 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 weโll 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 and ๐ and capital ๐น of ๐ฅ is defined by the integral between ๐ and ๐ฅ of ๐ of
๐ก with respect to ๐ก. Then ๐น prime of ๐ฅ is equal to ๐
of ๐ฅ for all values of ๐ฅ on the open interval between ๐ 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
weโve been given in the question does match the form of the fundamental theorem of
calculus with ๐ of ๐ฅ representing capital ๐น of ๐ฅ, the natural log of one plus ๐ก
to the power of five representing lowercase ๐ of ๐ก, the lower limit of our
integration three being the constant ๐, and of course, the upper limit being
๐ฅ. Given the forms match, we can
directly use the fundamental theorem of calculus to reach a result for ๐ prime of
๐ฅ, which here represents capital ๐น prime of ๐ฅ. We know the function lowercase ๐
of ๐ก and so to find lowercase ๐ of ๐ฅ, we simply replace the ๐กs by ๐ฅs. This means lowercase ๐ of ๐ฅ is
equal to the natural log of one plus ๐ฅ to the power of five. And in fact, weโve already reached
our answer for ๐ prime of ๐ฅ.