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

Fundamental Theorem of Calculus Part 1
Fundamental Theorem of Calculus Part 1

Video Transcript

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.

For this question, know that we’ve
been given a function 𝑔 of 𝑥, 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 𝑥. Here, this would be a mistake,
since the integral we’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 we’ll be
abbreviating to FTC. 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. Then 𝐹 prime of 𝑥 is equal to 𝑓
of 𝑥 for all values of 𝑥 on the open interval between 𝑎 and 𝑏.

This is an incredibly powerful
theorem and we can understand its meaning by applying it to our question. 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. 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. We know the function lowercase 𝑓
of 𝑡 and so to find lowercase 𝑓 of 𝑥, we simply replace the 𝑡s by 𝑥s. This means lowercase 𝑓 of 𝑥 is
equal to the natural log of one plus 𝑥 to the power of five. And in fact, we’ve already reached
our answer for 𝑔 prime of 𝑥.

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