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## Integral Calculus

### Course: Integral Calculus>Unit 1

Lesson 6: Fundamental theorem of calculus and accumulation functions

# Finding derivative with fundamental theorem of calculus

The Fundamental Theorem of Calculus tells us that the derivative of the definite integral from 𝘢 to 𝘹 of ƒ(𝑡)𝘥𝑡 is ƒ(𝘹), provided that ƒ is continuous. See how this can be used to evaluate the derivative of accumulation functions. Created by Sal Khan.

## Want to join the conversation?

• I don't really understand the role of the lower boundary. It seems to have no effect whatsoever to the result. What changes will take place when we change the lower boundary? •  The other aspect is that the lower boundary in these has always been a constant number (Pi) when taking the derivative of that boundary it is always zero and therefore irrelevant in the final answer. Check the fourth video in this sequence for more on this.
• I don't understand why he needs to put the derivative of x^2 in the final answer. Can someone please help me with this? •  It is because of the chain rule, as he mentioned. Actually, you ALWAYS have to put the d/dx (of the bound of the definite integral) in the answer. However, in the case where you just have x as the bound, the d/dx = 1. So, you are always putting that derivative in, but in the first example he showed the d/dx was just 1 and didn't affect the final answer.
• What he writes is NOT the second fundamental theorem of calculus. Rather, it is the first fundamental theorem, or the first "part," as some sources prefer. In fact, if you don't believe me, look at the video below (in the AP BC "Integration and accumulation of change" unit) entitled "The Fundamental Theorem of Calculus and definite integrals." In that video Sal labels the first and second theorems correctly, and in fact notes that sometimes they are referred to as "parts" of the Fundamental Theorem.

I know it's tough to redo videos, but shouldn't you have put one of those pop-ups you have in other videos, where you say things like "what Sal meant to say was ...". • Surely this is an improper integral, because cot^2(pi) is not defined. So my question is, does this still matter when applying the fundamental theorem of calculus? • If taking the derivative of an integral leaves you with f(x), then would it be right to think of integrals and derivatives as inverse functions? • At , why would we need to apply the chain rule? • I am having some problems conceptualizing the fundamental theorem of calculus,first thing i want to ask is this :when we take the derivative of an integral of some functions we get back to the original function because derivatives and integrals are inverse operators,right ? Second thing :after we take the derivative we get the original function at x but isnt f(t) the same function at x ,what is the distinction here ?i hope i explained my self ,sorry for my rude english.Thank you all • why is the chain rule used for Sal's second example? Shouldn't it just be cot^2(x^2)?   