How do you find the derivative of #y= arctan sqrt(x^2 -1) + arc csc x#?

Answer 1

0

Use # f ^(-1)f(a)=a#.

To make y real x in [-1, 1]. Befittingly,

let #x = sec theta in [-1, 1]#.

Now,

#y = arc tan (tan theta) + pi/2 - arc sec (sec theta)#,
using #arc csc x+arc sec x = pi/2#
#=theta +pi/2-theta#
#=pi/2# And so,
#y' = 0#.
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Answer 2

To find the derivative of (y = \arctan(\sqrt{x^2 - 1}) + \operatorname{arcsec}(x)), we'll use the chain rule and the derivative formulas for (\arctan(x)) and (\operatorname{arcsec}(x)).

First, recall the derivatives:

  • The derivative of (\arctan(u)) with respect to (x) is (\frac{1}{1+u^2} \cdot \frac{du}{dx}).
  • The derivative of (\operatorname{arcsec}(u)) with respect to (x) is (\frac{1}{|u|\sqrt{u^2-1}} \cdot \frac{du}{dx}), where (u = x) and thus (\frac{du}{dx} = 1).

Let's take it step by step:

  1. For (\arctan(\sqrt{x^2 - 1})):

    Let (u = \sqrt{x^2 - 1}), so we first find (\frac{du}{dx}).

    (u = (x^2 - 1)^{\frac{1}{2}})

    (\frac{du}{dx} = \frac{1}{2}(x^2 - 1)^{-\frac{1}{2}} \cdot 2x = \frac{x}{\sqrt{x^2 - 1}})

    Thus, the derivative of (\arctan(\sqrt{x^2 - 1})) is:

    (\frac{d}{dx}[\arctan(\sqrt{x^2 - 1})] = \frac{1}{1 + (\sqrt{x^2 - 1})^2} \cdot \frac{x}{\sqrt{x^2 - 1}})

    Simplifying inside the denominator of the first fraction:

    (1 + (\sqrt{x^2 - 1})^2 = 1 + x^2 - 1 = x^2)

    So, this part simplifies to:

    (\frac{x}{x^2\sqrt{x^2 - 1}} = \frac{1}{x\sqrt{x^2 - 1}})

  2. For (\operatorname{arcsec}(x)):

    Directly applying the derivative formula since (u = x):

    (\frac{d}{dx}[\operatorname{arcsec}(x)] = \frac{1}{|x|\sqrt{x^2 - 1}})

Combining both parts, the derivative of (y = \arctan(\sqrt{x^2 - 1}) + \operatorname{arcsec}(x)) is:

[\frac{dy}{dx} = \frac{1}{x\sqrt{x^2 - 1}} + \frac{1}{|x|\sqrt{x^2 - 1}}]

Given that both terms involve (\sqrt{x^2 - 1}) in the denominator, we keep them separate because of the absolute value in the second term, which affects the domain differently for (x > 0) and (x < 0).

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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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