How do you find the derivative of #y= arctan sqrt(x^2 -1) + arc csc x#?
0
To make y real x in [-1, 1]. Befittingly,
Now,
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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:
-
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}})
-
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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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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