In this video we're going to talk about the first part of the fundamental theorem of calculus so if g of x is equal to the definite integral of f of t from a to x then g prime of x is going to equal f of x so if g is the antiderivative of f then the derivative of g will equal f the derivative of the antiderivative will give you the original function another way in which you can express the second part of the i mean the first part of the fundamental theorem of calculus is you can express it this way the derivative of the integral from a to x of f of t dt is going to equal f of x and for the most part you're replacing t with x but that's not always the case as some examples will illustrate but that's the basic idea behind the first part of the fundamental theorem of calculus some textbooks may call it the second part of the fundamental theorem of calculus but regardless of what they call it the principle still remains the same now let's work on an example problem so let's find the derivative with respect to x of the integral from zero to x of the function the square root of t squared plus 4 dt now it's going to be difficult to integrate this function and then differentiate it so there has to be an easier way perhaps you realize that the answer is going to be f of x which is just the square root of x squared plus 4. but there's a process that you can employ to get the answer and we're going to go over that right now so f of t is going to be the...
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