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

Use the fundamental theorem of calculus to find the derivative of the function β„Ž(𝑒) = ∫_(4)^(𝑒) (√3𝑑/(4𝑑 + 2)) d𝑑.

02:15

Video Transcript

Use the fundamental theorem of calculus to find the derivative of the function β„Ž of 𝑒 is equal to the integral from four to 𝑒 of the square root of three 𝑑 divided by four 𝑑 plus two with respect to 𝑑.

The first thing we should notice is that we’re given a function β„Ž of 𝑒 which is defined by an integral. And the question is asking us to use the fundamental theorem of calculus to find the derivative of this function. We recall that the fundamental theorem of calculus states that if lowercase of 𝑓 is a continuous function on the closed interval π‘Ž to 𝑏 and capital 𝐹 of π‘₯ is equal to the integral from π‘Ž to π‘₯ of lowercase of 𝑓 of 𝑑 with respect to 𝑑. Then we can conclude that capital 𝐹 prime of π‘₯ is equal to lowercase of 𝑓 of π‘₯ whenever π‘₯ is in the open interval π‘Ž to 𝑏.

In our question, we have that β„Ž of 𝑒 is defined as an integral with 𝑒 as the upper limit of our integral. This means that the lowercase 𝑓 function in the fundamental theorem of calculus in our example will be the square root of three 𝑑 divided by four 𝑑 plus two. We remember that we require lowercase of 𝑓 to be continuous on the closed interval π‘Ž to 𝑏. We see that since lowercase 𝑓 of 𝑑 is the quotient to standard functions, the only time for lowercase 𝑓 of 𝑑 will be discontinuous is when the denominator is equal to zero. Which, in this case, occurs when 𝑑 is equal to negative a half. In particular, this gives us that lowercase 𝑓 of 𝑑 is continuous on the closed interval four to 𝑏 for any 𝑏 is greater than four.

Since we have shown that both prerequisites of the fundamental theorem of calculus are true for the function given to us in this question. We can use the fundamental theorem of calculus to conclude that β„Ž prime of 𝑒 is equal to a lowercase 𝑓 of 𝑒 whenever π‘₯ is greater than four. Finally, we know that 𝑓 of 𝑒 is equal to the square root of three 𝑒 divided by four 𝑒 plus two. Which means that we have shown, by using the fundamental theorem of calculus, the derivative of the function β„Ž of 𝑒, given to us in the question, is equal to the square root of three 𝑒 divided by four 𝑒 plus two.

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