How do you calculate the derivative of #intarctan t dt # from #[2,1/x]#?

Answer 1

# d/dxint_2^(1/x) arctan t dt = -(arctan (1/x))/x^2 #

# d/dxint_2^(1/x) arctan t dt = arctan (1/x) d/dx(1/x) # # = -(arctan (1/x))/x^2 #
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Answer 2

Use the Fundamental Theorem of Calculus, Part 1 and the chain rule.

Fundamental Theorem of Calculus, Part 1 tells us that if #f# is continuous on interval #[a,b]#, if #g# is defined by
#g(x) = int_a^x f(t) dt" "# for #x in [a,b]#
then (#g# is continuous on #[a,b]#) and (#g# is differentiable on #(a,b)#)
and #g'(x) = f(x)# (which is what we need here).

In this problem we have a composition, so we need the chain rule:

#g(x) = int_2^u arctan t dt# with #u = 1/x#

So the chain rule gives us:

#g'(x) = arctanu (du)/dx#

In this case:

#g'(x) = arctan(1/x) (-1)/x^2#
# = - arctan(1/x)/x^2#
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Answer 3

To calculate the derivative of ∫arctan(t) dt from [2, 1/x], where x is a variable:

  1. First, find the antiderivative of arctan(t), which is ∫arctan(t) dt = t * arctan(t) - (1/2) * ln(1 + t^2) + C, where C is the constant of integration.
  2. Evaluate the antiderivative at the upper bound (1/x) and subtract from it the value of the antiderivative at the lower bound (2).

So, the derivative of ∫arctan(t) dt from [2, 1/x] is:

d/dx [∫arctan(t) dt from 2 to 1/x] = [1/x * arctan(1/x) - (1/2) * ln(1 + (1/x)^2)] - [2 * arctan(2) - (1/2) * ln(1 + 2^2)]

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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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