How do you find the derivative of #g(x) = int_3^x e^(4t²-t) dt# ?

Answer 1

Answering this question is one of the main jobs of the Fundamental Theorem of Calculus, Part I. (Its other main job is to help us prove the Fundamental Theorem of Calculus, Part II.)

FTC Part I: If #f# is continuous on #[a,b]# and #g(x)# is defined by:
#color(white)"sssssssss"# #g(x) = int_a^x f(t)dt# for #x in [a,b]#, then
#color(white)"ss"# #g'(x) = f(x)#
So, for #g(x) = int_3^x e^(4t²-t) dt#, we get
#g'(x) = e^(4x²-x) #.
Yes. That's it. Yes, really. Just change the #t#'s to #x#'s and move on to the next question.
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Answer 2

To find the derivative of ( g(x) = \int_{3}^{x} e^{4t^2 - t} , dt ), we can apply the Fundamental Theorem of Calculus. According to this theorem, if ( g(x) ) is defined as the integral from a constant (in this case, 3) to a variable (x) of a function (in this case, ( e^{4t^2 - t} )), then the derivative of ( g(x) ) with respect to ( x ) is simply the integrand evaluated at ( x ). Therefore, the derivative of ( g(x) ) is ( e^{4x^2 - x} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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